Index: Content Tree

The tree index reflects the "common-thread" structure of BoP:

Branches (2839)
- Branch: Introduction to the Axiomatic Method
- Branch: Logic (92)
  - Part: Historical Development of Logic
  - Part: Basic Concepts of Logic (18)
    - Definition: Strings (words) over an Alphabet (2)
      - Example: Examples of Strings over Alphabets (related to Definition: Strings (words) over an Alphabet)
      - Corollary: Algebraic Structure of Strings over an Alphabet (related to Definition: Strings (words) over an Alphabet) (1)
        
        Proof: (related to Corollary: Algebraic Structure of Strings over an Alphabet)
    - Definition: Language (3)
      - Definition: Concatenation of Languages
      - Example: Examples of Languages (related to Definition: Language)
      - Definition: Iteration of Languages, Kleene Star, Kleene Plus
    - Definition: Grammar (Syntax) (1)
      - Example: Examples of Syntax (related to Definition: Grammar (Syntax))
    - Definition: Formal Languages Generated From a Grammar
    - Chapter: From the Syntax to the Semantics of Formal Languages (4)
      - Axiom: Bivalence of Truth
      - Definition: Set of Truth Values (True and False)
      - Definition: Domain of Discourse
      - Definition: Interpretation of Strings of a Formal Language and Their Truth Function
    - Definition: Satisfaction Relation, Model, Tautology, Contradiction
    - Definition: Semantics of a Formal Language
    - Chapter: The Proving Machine - an Automation of Logical Reasoning (4)
      - Definition: Axioms
      - Definition: Rules of Inference
      - Definition: Logical Calculus
      - Definition: Proofs and Theorems in a Logical Calculus
    - Definition: Derivability Property
  - Part: Proof Theory (8)
    - Chapter: Putting it All Together - Syntax and Semantics of a Logical Calculus (3)
      - Definition: Soundness and Completeness of a Logical Calculus
      - Definition: Consistency and Negation-Completeness of a Logical Calculus
      - Definition: Negation of a String
    - Chapter: Classifying the Logical Calculi - Variables, Formulas, Predicates and Signatures (5)
      - Definition: Variable in a Logical Calculus (1)
        
        Example: Examples of Variables in a Logical Calculus (related to Definition: Variable in a Logical Calculus)
      - Definition: Quantifier, Bound Variables, Free Variables (1)
        
        Example: Examples of Quantifiers in a Logical Calculus (related to Definition: Quantifier, Bound Variables, Free Variables)
      - Definition: Function, Arity and Constant (1)
        
        Example: Examples of Functions in a Logical Calculus (related to Definition: Function, Arity and Constant)
      - Definition: Predicate of a Logical Calculus (1)
        
        Example: Examples of Predicates in a Logical Calculus (related to Definition: Predicate of a Logical Calculus)
      - Definition: Signature
  - Part: Propositional Logic (35)
    - Definition: Signature of Propositional Logic - PL0
    - Definition: Syntax of PL0 - Propositions as Boolean Terms
    - Definition: Interpretation of Propositions - the Law of the Excluded Middle (2)
      - Example: Examples of Propositions With a More Complex Syntax (related to Definition: Interpretation of Propositions - the Law of the Excluded Middle)
      - Definition: Paradox (1)
        
        Example: Examples of Paradoxes (related to Definition: Paradox)
    - Definition: Semantics of PL0 (15)
      - Definition: `$k$`-nary Connectives, Prime and Compound Propositions
      - Lemma: Boolean Function (2)
        
        Example: Examples of Boolean Functions (related to Lemma: Boolean Function)
        
        Proof: (related to Lemma: Boolean Function)
      - Definition: Truth Table
      - Definition: Negation
      - Definition: Conjunction (2)
        
        Corollary: Commutativity of Conjunction (related to Definition: Conjunction) (1)
        
        Proof: (related to Corollary: Commutativity of Conjunction)
        
        Proposition: Associativity of Conjunction (1)
        
        Proof: (related to Proposition: Associativity of Conjunction)
      - Definition: Disjunction (3)
        
        Corollary: Commutativity of Disjunction (related to Definition: Disjunction) (1)
        
        Proof: (related to Corollary: Commutativity of Disjunction)
        
        Proposition: Associativity of Disjunction (1)
        
        Proof: (related to Proposition: Associativity of Disjunction)
        
        Definition: Exclusive Disjunction
      - Definition: Implication (4)
        
        Lemma: Implication as a Disjunction (1)
        
        Proof: (related to Lemma: Implication as a Disjunction)
        
        Lemma: Negation of an Implication (2)
        
        Proof: (related to Lemma: Negation of an Implication)
        
        Example: Examples of Correct and Incorrect Negations of Implications (related to Lemma: Negation of an Implication)
        
        Definition: Contrapositive
      - Definition: Equivalence (1)
        
        Corollary: Commutativity of Equivalence (related to Definition: Equivalence) (1)
        
        Proof: (related to Corollary: Commutativity of Equivalence)
    - Chapter: Equivalent Transformations in Propositional Logic (5)
      - Lemma: Every Proposition Implies Itself (1)
        
        Proof: (related to Lemma: Every Proposition Implies Itself)
      - Lemma: It is true that something can be (either) true or false (1)
        
        Proof: (related to Lemma: It is true that something can be (either) true or false)
      - Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition (1)
        
        Proof: (related to Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition)
      - Lemma: De Morgan's Laws (Logic) (1)
        
        Proof: (related to Lemma: De Morgan's Laws (Logic))
      - Lemma: Distributivity of Conjunction and Disjunction (1)
        
        Proof: (related to Lemma: Distributivity of Conjunction and Disjunction)
    - Chapter: Contradictory Propositions in Propositional Logic (2)
      - Lemma: A proposition cannot be both, true and false (1)
        
        Proof: (related to Lemma: A proposition cannot be both, true and false)
      - Lemma: A proposition cannot be equivalent to its negation (1)
        
        Proof: (related to Lemma: A proposition cannot be equivalent to its negation)
    - Chapter: Normal Forms in `$PL0$` (8)
    - Definition: Boolean Algebra (1)
      - Lemma: Boolean Algebra of Propositional Logic (1)
        
        Proof: (related to Lemma: Boolean Algebra of Propositional Logic)
  - Part: PL1 - First Order Predicate Logic (3)
    - Definition: Terms in Predicate Logic
    - Definition: Atomic Formulae in Predicate Logic
    - Chapter: Peano Arithmetic
  - Part: Higher-Order Logics
  - Part: Gödel's Incompleteness Theorems
  - Part: Methods of Mathematical Proving (24)
    - Definition: Logical Arguments
    - Example: Examples of Logical Arguments (related to Part: Methods of Mathematical Proving)
    - Chapter: Invalid Logical Arguments (5)
      - Explanation: Inductive Reasoning (related to Chapter: Invalid Logical Arguments)
      - Lemma: Mixing-up the Inclusive and Exclusive Disjunction (1)
        
        Proof: (related to Lemma: Mixing-up the Inclusive and Exclusive Disjunction)
      - Lemma: Mixing-up the Sufficient and Necessary Conditions (1)
        
        Proof: (related to Lemma: Mixing-up the Sufficient and Necessary Conditions)
      - Lemma: Affirming the Consequent of an Implication (1)
        
        Proof: (related to Lemma: Affirming the Consequent of an Implication)
      - Lemma: Denying the Antecedent of an Implication (1)
        
        Proof: (related to Lemma: Denying the Antecedent of an Implication)
    - Chapter: Logical Arguments Used in Mathematical Proofs (11)
      - Explanation: Deductive Reasoning (related to Chapter: Logical Arguments Used in Mathematical Proofs)
      - Lemma: A Criterion for Valid Logical Arguments (1)
        
        Proof: (related to Lemma: A Criterion for Valid Logical Arguments)
      - Lemma: Modus Ponens (1)
        
        Proof: (related to Lemma: Modus Ponens)
      - Lemma: Modus Tollens (1)
        
        Proof: (related to Lemma: Modus Tollens)
      - Lemma: Hypothetical Syllogism (1)
        
        Proof: (related to Lemma: Hypothetical Syllogism)
      - Lemma: Disjunctive Syllogism (1)
        
        Proof: (related to Lemma: Disjunctive Syllogism)
      - Lemma: The Proving Principle by Contradiction (1)
        
        Proof: (related to Lemma: The Proving Principle by Contradiction)
      - Lemma: The Proving Principle By Contraposition, Contrapositive (1)
        
        Proof: (related to Lemma: The Proving Principle By Contraposition, Contrapositive)
      - Lemma: The Proving Principle by Complete Induction (2)
        
        Proof: By Induction (related to Lemma: The Proving Principle by Complete Induction)
        
        Proof: (related to Lemma: The Proving Principle by Complete Induction)
      - Lemma: The Proving Principle by Transfinite Induction (1)
        
        Proof: (related to Lemma: The Proving Principle by Transfinite Induction)
    - Explanation: When is something "well-defined" in mathematics? (related to Part: Methods of Mathematical Proving)
    - Explanation: Good Practices for Writing Mathematical Proofs (related to Part: Methods of Mathematical Proving)
    - Explanation: Correct Negation of Statements (related to Part: Methods of Mathematical Proving)
    - Motivation: What does not have to be proved in mathematics? (related to Part: Methods of Mathematical Proving)
    - Explanation: What does WLOG mean? (related to Part: Methods of Mathematical Proving)
    - Example: Existence and Uniqueness of Solutions (related to Part: Methods of Mathematical Proving)
  - Part: Solving Strategies and Sample Solutions to Problems in Logic
- Branch: Set Theory (153)
  - Part: Historical Development of Set Theory
  - Part: Basics about Sets (20)
    - Definition: Set, Set Element, Empty Set (1)
      - Explanation: Possibilities to Describe Sets, Venn-Diagrams, List, and Set-Builder Notations (related to Definition: Set, Set Element, Empty Set)
    - Definition: Universal Set
    - Definition: Subset and Superset
    - Definition: Power Set
    - Chapter: Set Operations (12)
      - Definition: Set Union (3)
        
        Proposition: Sets are Subsets of Their Union (1)
        
        Proof: (related to Proposition: Sets are Subsets of Their Union)
        
        Proposition: Set Union is Commutative (1)
        
        Proof: (related to Proposition: Set Union is Commutative)
        
        Proposition: Set Union is Associative (1)
        
        Proof: (related to Proposition: Set Union is Associative)
      - Definition: Set Intersection (4)
        
        Definition: Disjoint Sets
        
        Proposition: Intersection of a Set With Another Set is Subset of This Set (1)
        
        Proof: (related to Proposition: Intersection of a Set With Another Set is Subset of This Set)
        
        Proposition: Set Intersection is Commutative (1)
        
        Proof: (related to Proposition: Set Intersection is Commutative)
        
        Proposition: Set Intersection is Associative (1)
        
        Proof: (related to Proposition: Set Intersection is Associative)
      - Definition: Set Complement
      - Definition: Set Difference
      - Proposition: De Morgan's Laws (Sets) (1)
        
        Proof: (related to Proposition: De Morgan's Laws (Sets))
      - Proposition: Distributivity Laws For Sets (1)
        
        Proof: (related to Proposition: Distributivity Laws For Sets)
      - Proposition: Sets and Their Complements (1)
        
        Proof: (related to Proposition: Sets and Their Complements)
    - Definition: Index Set and Set Family
    - Definition: Mutually Disjoint Sets
    - Definition: Set Partition
    - Definition: Generalized Union of Sets
  - Motivation: Russell's Paradox (related to Branch: Set Theory)
  - Part: Zermelo-Fraenkel Set Theory (21)
    - Axiom: Zermelo-Fraenkel Axioms (21)
      - Axiom: Axiom of Existence
      - Axiom: Axiom of Empty Set (1)
        
        Corollary: Uniqueness of the Empty Set (related to Axiom: Axiom of Empty Set) (1)
        
        Proof: (related to Corollary: Uniqueness of the Empty Set)
      - Axiom: Axiom of Extensionality (1)
        
        Explanation: What does the Extensionality Principle mean? (related to Axiom: Axiom of Extensionality)
      - Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension) (8)
        
        Corollary: Justification of the Set-Builder Notation (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of the Set-Builder Notation)
        
        Corollary: Justification of Subsets and Supersets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of Subsets and Supersets)
        
        Corollary: Equality of Sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Equality of Sets)
        
        Explanation: How the Axiom of Separation Avoids Russel's Paradox (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
        
        Corollary: There is no set of all sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: There is no set of all sets)
        
        Corollary: Justification of Set Intersection (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of Set Intersection)
        
        Corollary: Justification of the Difference (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Justification of the Difference)
        
        Corollary: Set Difference and Set Complement are the Same Concepts (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
        
        Proof: (related to Corollary: Set Difference and Set Complement are the Same Concepts)
      - Axiom: Axiom of Pairing
      - Axiom: Axiom of Union (1)
        
        Corollary: Justification of Set Union (related to Axiom: Axiom of Union) (1)
        
        Proof: (related to Corollary: Justification of Set Union)
      - Axiom: Axiom of Power Set (2)
        
        Corollary: Justification of Power Set (related to Axiom: Axiom of Power Set) (1)
        
        Proof: (related to Corollary: Justification of Power Set)
        
        Definition: Singleton
      - Axiom: Axiom of Foundation (1)
        
        Corollary: Circular References Of Self-Contained Sets Are Forbidden (related to Axiom: Axiom of Foundation) (1)
        
        Proof: By Induction (related to Corollary: Circular References Of Self-Contained Sets Are Forbidden)
      - Axiom: Axiom of Infinity (3)
        
        Definition: Inductive Set
        
        Definition: Minimal Inductive Set
        
        Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets (related to Axiom: Axiom of Infinity) (1)
        
        Proof: (related to Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets)
      - Axiom: Axiom of Replacement (Schema)
      - Axiom: Axiom of Choice
  - Part: Relations (60)
    - Definition: Ordered Pair, n-Tuple (1)
      - Proposition: Set-Theoretical Meaning of Ordered Tuples (1)
        
        Proof: By Induction (related to Proposition: Set-Theoretical Meaning of Ordered Tuples)
    - Motivation: Usage of Ordered Tuples In Other Mathematical Disciplines (related to Part: Relations)
    - Definition: Cartesian Product
    - Definition: Relation (1)
      - Example: Examples of Relations (related to Definition: Relation)
    - Chapter: Binary Relations and Their Properties (7)
      - Definition: Inverse Relation
      - Explanation: Representations of Binary Relations (related to Chapter: Binary Relations and Their Properties)
      - Definition: Total and Unique Binary Relations
      - Definition: Composition of Binary Relations (2)
        
        Lemma: Composition of Relations Preserves Their Right-Uniqueness Property (1)
        
        Proof: (related to Lemma: Composition of Relations Preserves Their Right-Uniqueness Property)
        
        Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property (1)
        
        Proof: (related to Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property)
      - Definition: Reflexive, Symmetric and Transitive Binary Relations
      - Definition: Irreflexive, Asymmetric and Antisymmetric Binary Relations
    - Definition: Equivalence Relation (7)
      - Motivation: Significance of Equivalence Relations (related to Definition: Equivalence Relation)
      - Example: Examples of Equivalence Relations (related to Definition: Equivalence Relation)
      - Proposition: The Equality of Sets Is an Equivalence Relation (1)
        
        Proof: (related to Proposition: The Equality of Sets Is an Equivalence Relation)
      - Definition: Equivalence Class
      - Definition: Quotient Set, Partition
      - Definition: Complete System of Representatives
      - Definition: Canonical Projection
    - Chapter: Functions (Maps) (25)
      - Definition: Partial and Total Maps (Functions)
      - Explanation: Some Remarks on Functions (related to Chapter: Functions (Maps))
      - Definition: Graph of a Function (1)
        
        Example: Graph of a Real-Valued Function with One Variable (related to Definition: Graph of a Function)
      - Section: Important Properties of Functions (6)
        
        Definition: Injective Function
        
        Definition: Surjective Function
        
        Definition: Bijective Function
        
        Proposition: Characterization of Bijective Functions (1)
        
        Proof: (related to Proposition: Characterization of Bijective Functions)
        
        Definition: Invertible Functions, Inverse Functions
        
        Proposition: Functions Constitute Equivalence Relations (1)
        
        Proof: (related to Proposition: Functions Constitute Equivalence Relations)
      - Section: Common Types of Functions (6)
        
        Definition: Identity Function
        
        Definition: Indicator (Characteristic) Function, Carrier
        
        Definition: Constant Function
        
        Definition: Embedding, Inclusion Map
        
        Definition: Restriction
        
        Definition: Fixed Point, Fixed Point Property
      - Lemma: Behavior of Functions with Set Operations (2)
        
        Proof: (related to Lemma: Behavior of Functions with Set Operations)
        
        Explanation: How Functions interact with Set Operations (related to Lemma: Behavior of Functions with Set Operations)
      - Lemma: Composition of Functions (7)
        
        Proof: (related to Lemma: Composition of Functions)
        
        Proposition: Composition of Functions is Associative (1)
        
        Proof: (related to Proposition: Composition of Functions is Associative)
        
        Proposition: Composition of Surjective Functions is Surjective (1)
        
        Proof: (related to Proposition: Composition of Surjective Functions is Surjective)
        
        Proposition: Composition of Injective Functions is Injective (1)
        
        Proof: (related to Proposition: Composition of Injective Functions is Injective)
        
        Proposition: Composition of Bijective Functions is Bijective (1)
        
        Proof: (related to Proposition: Composition of Bijective Functions is Bijective)
        
        Proposition: Injective, Surjective and Bijective Compositions (1)
        
        Proof: (related to Proposition: Injective, Surjective and Bijective Compositions)
        
        Proposition: The Inverse Of a Composition (1)
        
        Proof: (related to Proposition: The Inverse Of a Composition)
      - Definition: Zero of a Function
    - Chapter: Order Relations (17)
      - Definition: Preorder, Partial Order and Poset
      - Definition: Total Order and Chain
      - Definition: Comparing the Elements of Posets and Chains
      - Definition: Strict Total Order, Strictly-ordered Set
      - Lemma: Comparing the Elements of Strictly Ordered Sets (1)
        
        Proof: (related to Lemma: Comparing the Elements of Strictly Ordered Sets)
      - Explanation: Summary of Different Order Relations (related to Chapter: Order Relations)
      - Explanation: Hasse Diagram (related to Chapter: Order Relations)
      - Definition: Special Elements of Ordered Sets
      - Explanation: Notes on Special Elements of Posets (related to Chapter: Order Relations)
      - Definition: Bounded Subsets of Ordered Sets
      - Definition: Bounded Subsets of Unordered Sets
      - Lemma: Zorn's Lemma (1)
        
        Proof: (related to Lemma: Zorn's Lemma)
      - Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice (1)
        
        Proof: (related to Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice)
      - Definition: Well-order, Well-ordered Set (3)
        
        Explanation: A Note on Well-ordered Sets (related to Definition: Well-order, Well-ordered Set)
        
        Proposition: Well-ordered Sets are Chains (1)
        
        Proof: (related to Proposition: Well-ordered Sets are Chains)
        
        Proposition: Finite Chains are Well-ordered (1)
        
        Proof: (related to Proposition: Finite Chains are Well-ordered)
      - Definition: Order Embedding
  - Part: Ordinal Numbers (26)
    - Motivation: Is "Being a Set Element" ("`$\in$`") a Relation? (related to Part: Ordinal Numbers)
    - Proposition: Transitive Recursion (1)
      - Proof: (related to Proposition: Transitive Recursion)
    - Definition: Contained Relation "`$\in_X$`"
    - Proposition: Contained Relation is a Strict Order (1)
      - Proof: (related to Proposition: Contained Relation is a Strict Order)
    - Definition: Transitive Set (2)
      - Corollary: Properties of Transitive Sets (related to Definition: Transitive Set) (1)
        
        Proof: (related to Corollary: Properties of Transitive Sets)
      - Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull (1)
        
        Proof: (related to Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull)
    - Definition: Extensional Relation (3)
      - Explanation: Examples of Extensional Relations (related to Definition: Extensional Relation) (3)
        
        Proposition: The Contained Relation is Extensional (1)
        
        Proof: (related to Proposition: The Contained Relation is Extensional)
        
        Proposition: Partial Orders are Extensional (1)
        
        Proof: By Contradiction (related to Proposition: Partial Orders are Extensional)
        
        Proposition: Strict Orders are Extensional (1)
        
        Proof: By Contradiction (related to Proposition: Strict Orders are Extensional)
    - Definition: Well-founded Relation (1)
      - Example: Examples of Well-founded Relations (related to Definition: Well-founded Relation)
    - Definition: Mostowski Function and Collapse (6)
    - Definition: Ordinal Number (4)
      - Proposition: Equivalent Notions of Ordinals (1)
        
        Proof: (related to Proposition: Equivalent Notions of Ordinals)
      - Proposition: Ordinals Are Downward Closed (1)
        
        Proof: (related to Proposition: Ordinals Are Downward Closed)
      - Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals (1)
        
        Proof: (related to Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals)
      - Theorem: Trichotomy of Ordinals (1)
        
        Proof: (related to Theorem: Trichotomy of Ordinals)
    - Lemma: Properties of Ordinal Numbers (1)
      - Proof: (related to Lemma: Properties of Ordinal Numbers)
    - Lemma: Successor of Ordinal (1)
      - Proof: (related to Lemma: Successor of Ordinal)
    - Definition: Limit Ordinal
    - Explanation: Building the Successors of Ordinal Numbers (related to Part: Ordinal Numbers)
    - Motivation: Burali-Forti Paradox (related to Part: Ordinal Numbers)
    - Definition: The Class of all Ordinals `$\Omega$`
  - Part: Cardinal Numbers (23)
    - Definition: Equipotent Sets
    - Proposition: Cardinal Number (1)
      - Proof: (related to Proposition: Cardinal Number)
    - Definition: Finite Set, Infinite Set
    - Definition: Comparison of Cardinal Numbers
    - Chapter: Can Cardinals be Ordered?
    - Theorem: Schröder-Bernstein Theorem (1)
      - Proof: (related to Theorem: Schröder-Bernstein Theorem)
    - Chapter: Simple Facts Regarding Cardinals (5)
      - Proposition: Subsets of Finite Sets (1)
        
        Proof: (related to Proposition: Subsets of Finite Sets)
      - Theorem: Distinction Between Finite and Infinite Sets Using Subsets (1)
        
        Proof: (related to Theorem: Distinction Between Finite and Infinite Sets Using Subsets)
      - Proposition: More Characterizations of Finite Sets (1)
        
        Proof: (related to Proposition: More Characterizations of Finite Sets)
      - Lemma: Finite Cardinal Numbers and Set Operations (1)
        
        Proof: (related to Lemma: Finite Cardinal Numbers and Set Operations)
      - Proposition: Counting the Set's Elements Using Its Partition (1)
        
        Proof: (related to Proposition: Counting the Set's Elements Using Its Partition)
    - Motivation: Cantor's Astonishing Discoveries Regarding the Cardinals of Infinite Sets (related to Part: Cardinal Numbers)
    - Explanation: Transitive Set and Countability - Natural Numbers Have the Smallest Infinite Cardinality (related to Part: Cardinal Numbers)
    - Definition: Countable Set, Uncountable Set
    - Proposition: Union of Countably Many Countable Sets (2)
      - Corollary: Cartesian Products of Countable Sets Is Countable (related to Proposition: Union of Countably Many Countable Sets) (1)
        
        Proof: By Induction (related to Corollary: Cartesian Products of Countable Sets Is Countable)
      - Proof: (related to Proposition: Union of Countably Many Countable Sets)
    - Proposition: Cardinals of a Set and Its Power Set (1)
      - Proof: (related to Proposition: Cardinals of a Set and Its Power Set)
    - Proposition: Subset of a Countable Set is Countable (1)
      - Proof: (related to Proposition: Subset of a Countable Set is Countable)
    - Proposition: Rational Numbers are Countable (1)
      - Proof: Uncountable Set (related to Proposition: Rational Numbers are Countable)
    - Proposition: Real Numbers are Uncountable (2)
      - Proof: What does countability mean? (related to Proposition: Real Numbers are Uncountable)
      - Corollary: Irrational Numbers are Uncountable (related to Proposition: Real Numbers are Uncountable) (1)
        
        Proof: (related to Corollary: Irrational Numbers are Uncountable)
    - Proposition: Uncountable and Countable Subsets of Natural Numbers (1)
      - Proof: (related to Proposition: Uncountable and Countable Subsets of Natural Numbers)
    - Chapter: Continuum Hypothesis
  - Part: Solving Strategies and Sample Solutions to Problems in Set Theory (1)
    - Problem: To Construct a Partition of a Given Set (1)
      - Solution: (related to Problem: To Construct a Partition of a Given Set)
- Branch: Number Systems and Arithmetics (223)
  - Part: Natural Numbers (30)
    - Definition: Set-theoretic Definitions of Natural Numbers (15)
      - Proposition: Addition Of Natural Numbers (7)
        
        Proof: (related to Proposition: Addition Of Natural Numbers)
        
        Proposition: Addition Of Natural Numbers Is Associative (1)
        
        Proof: By Induction (related to Proposition: Addition Of Natural Numbers Is Associative)
        
        Proposition: Addition of Natural Numbers Is Commutative (1)
        
        Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Commutative)
        
        Proposition: Addition of Natural Numbers Is Cancellative (2)
        
        Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Cancellative)
        
        Corollary: Contraposition of Cancellative Law for Adding Natural Numbers (related to Proposition: Addition of Natural Numbers Is Cancellative) (1)
        
        Proof: (related to Corollary: Contraposition of Cancellative Law for Adding Natural Numbers)
        
        Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) (related to Proposition: Addition Of Natural Numbers) (1)
        
        Proof: (related to Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers))
        
        Proposition: Uniqueness of Natural Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Natural Zero)
      - Definition: Multiplication of Natural Numbers (6)
        
        Proposition: Multiplication of Natural Numbers Is Associative (1)
        
        Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Associative)
        
        Proposition: Multiplication of Natural Numbers is Commutative (1)
        
        Proof: By Induction (related to Proposition: Multiplication of Natural Numbers is Commutative)
        
        Proposition: Multiplication of Natural Numbers Is Cancellative (2)
        
        Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers)
        
        Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) (related to Definition: Multiplication of Natural Numbers) (1)
        
        Proof: (related to Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers))
        
        Proposition: Uniqueness Of Natural One (1)
        
        Proof: (related to Proposition: Uniqueness Of Natural One)
      - Proposition: Uniqueness Of Predecessors Of Natural Numbers (1)
        
        Proof: (related to Proposition: Uniqueness Of Predecessors Of Natural Numbers)
      - Proposition: Inequality of Natural Numbers and Their Successors (1)
        
        Proof: (related to Proposition: Inequality of Natural Numbers and Their Successors)
    - Axiom: Peano Axioms (2)
      - Definition: Set of Natural Numbers (Peano)
      - Explanation: Why do the Peano axioms define natural numbers? (related to Axiom: Peano Axioms)
    - Proposition: Algebraic Structure Of Natural Numbers Together With Addition (1)
      - Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Addition)
    - Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication)
    - Proposition: Distributivity Law For Natural Numbers (1)
      - Proof: By Induction (related to Proposition: Distributivity Law For Natural Numbers)
    - Chapter: Order (10)
      - Definition: Set-theoretic Definition of Order Relation for Natural Numbers
      - Definition: Order Relation for Natural Numbers (7)
        
        Proposition: Transitivity of the Order Relation of Natural Numbers (1)
        
        Proof: (related to Proposition: Transitivity of the Order Relation of Natural Numbers)
        
        Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities (1)
        
        Proof: (related to Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities)
        
        Proposition: Order Relation for Natural Numbers, Revised (1)
        
        Proof: (related to Proposition: Order Relation for Natural Numbers, Revised)
        
        Proposition: Every Natural Number Is Greater or Equal Zero (1)
        
        Proof: (related to Proposition: Every Natural Number Is Greater or Equal Zero)
        
        Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation (1)
        
        Proof: (related to Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation)
        
        Proposition: Comparing Natural Numbers Using the Concept of Addition (2)
        
        Proof: (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
        
        Corollary: Order Relation for Natural Numbers is Strict Total (related to Proposition: Comparing Natural Numbers Using the Concept of Addition) (1)
        
        Proof: (related to Corollary: Order Relation for Natural Numbers is Strict Total)
      - Proposition: Well-Ordering Principle of Natural Numbers (1)
        
        Proof: (related to Proposition: Well-Ordering Principle of Natural Numbers)
      - Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers (1)
        
        Proof: (related to Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers)
  - Part: Integers (25)
    - Proposition: Definition of Integers (20)
      - Proof: (related to Proposition: Definition of Integers)
      - Proposition: Addition of Integers (9)
        
        Proof: (related to Proposition: Addition of Integers)
        
        Proposition: Addition of Integers Is Associative (2)
        
        Proof: (related to Proposition: Addition of Integers Is Associative)
        
        Proof: (related to Proposition: Addition of Integers Is Associative)
        
        Proposition: Addition of Integers Is Commutative (1)
        
        Proof: (related to Proposition: Addition of Integers Is Commutative)
        
        Proposition: Addition of Integers Is Cancellative (2)
        
        Proof: (related to Proposition: Addition of Integers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Adding Integers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Integers)
        
        Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers) (1)
        
        Proof: (related to Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers))
        
        Proposition: Uniqueness of Integer Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Integer Zero)
        
        Proposition: Existence of Inverse Integers With Respect to Addition (1)
        
        Proof: (related to Proposition: Existence of Inverse Integers With Respect to Addition)
      - Definition: Subtraction of Integers
      - Proposition: Multiplication of Integers (8)
        
        Proof: (related to Proposition: Multiplication of Integers)
        
        Proposition: Multiplication of Integers Is Associative (1)
        
        Proof: (related to Proposition: Multiplication of Integers Is Associative)
        
        Proposition: Multiplication of Integers Is Commutative (1)
        
        Proof: (related to Proposition: Multiplication of Integers Is Commutative)
        
        Proposition: Multiplication of Integers Is Cancellative (2)
        
        Proof: (related to Proposition: Multiplication of Integers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Multiplying Integers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Integers)
        
        Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers) (1)
        
        Proof: (related to Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers))
        
        Proposition: Uniqueness of Integer One (1)
        
        Proof: (related to Proposition: Uniqueness of Integer One)
        
        Proposition: Multiplying Negative and Positive Integers (1)
        
        Proof: (related to Proposition: Multiplying Negative and Positive Integers)
      - Proposition: Distributivity Law For Integers (1)
        
        Proof: (related to Proposition: Distributivity Law For Integers)
    - Proposition: Algebraic Structure of Integers Together with Addition (2)
      - Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
      - Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
    - Proposition: Algebraic Structure of Integers Together with Addition and Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition and Multiplication)
    - Definition: Order Relation for Integers - Positive and Negative Integers (2)
      - Proposition: Order Relation for Integers is Strict Total (1)
        
        Proof: (related to Proposition: Order Relation for Integers is Strict Total)
      - Definition: Absolute Value of Integers
  - Part: Rational Numbers (28)
    - Proposition: Definition of Rational Numbers (22)
      - Proof: (related to Proposition: Definition of Rational Numbers)
      - Proposition: Addition Of Rational Numbers (9)
        
        Proof: (related to Proposition: Addition Of Rational Numbers)
        
        Proposition: Addition of Rational Numbers Is Associative (1)
        
        Proof: (related to Proposition: Addition of Rational Numbers Is Associative)
        
        Proposition: Addition of Rational Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Addition of Rational Numbers Is Commutative)
        
        Proposition: Addition of Rational Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Addition of Rational Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Adding Rational Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Rational Numbers)
        
        Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers) (1)
        
        Proof: (related to Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers))
        
        Proposition: Existence of Inverse Rational Numbers With Respect to Addition (2)
        
        Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Addition)
        
        Proposition: Position of Minus Sign in Rational Numbers Representations (1)
        
        Proof: (related to Proposition: Position of Minus Sign in Rational Numbers Representations)
        
        Proposition: Uniqueness of Rational Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Rational Zero)
      - Proposition: Multiplication Of Rational Numbers (10)
        
        Proof: (related to Proposition: Multiplication Of Rational Numbers)
        
        Proposition: Multiplication of Rational Numbers Is Associative (1)
        
        Proof: (related to Proposition: Multiplication of Rational Numbers Is Associative)
        
        Proposition: Multiplication Of Rational Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Multiplication Of Rational Numbers Is Commutative)
        
        Proposition: Multiplication Of Rational Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Multiplication Of Rational Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers)
        
        Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers) (1)
        
        Proof: (related to Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers))
        
        Proposition: Uniqueness Of Rational One (1)
        
        Proof: (related to Proposition: Uniqueness Of Rational One)
        
        Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication)
        
        Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication)
        
        Proposition: Multiplying Negative and Positive Rational Numbers (1)
        
        Proof: (related to Proposition: Multiplying Negative and Positive Rational Numbers)
      - Definition: Subtraction of Rational Numbers
      - Proposition: Distributivity Law For Rational Numbers (1)
        
        Proof: (related to Proposition: Distributivity Law For Rational Numbers)
    - Proposition: Algebraic Structure of Rational Numbers Together with Addition (1)
      - Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition)
    - Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication)
    - Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication (2)
      - Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
      - Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
    - Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers (1)
      - Proposition: Order Relation for Rational Numbers is Strict Total (1)
        
        Proof: (related to Proposition: Order Relation for Rational Numbers is Strict Total)
    - Definition: Absolute Value of Rational Numbers (1)
      - Corollary: The absolute value makes the set of rational numbers a metric space. (related to Definition: Absolute Value of Rational Numbers) (1)
        
        Proof: (related to Corollary: The absolute value makes the set of rational numbers a metric space.)
  - Part: Irrational Numbers (3)
    - Definition: Definition of Irrational Numbers
    - Proposition: Discovery of Irrational Numbers (2)
      - Proof: (related to Proposition: Discovery of Irrational Numbers)
      - Proof: (related to Proposition: Discovery of Irrational Numbers)
  - Part: Real Numbers (73)
    - Chapter: Real Numbers As Limits Of Rational Numbers (22)
    - Proposition: Definition of Real Numbers (39)
      - Proof: (related to Proposition: Definition of Real Numbers)
      - Proposition: Addition of Real Numbers (14)
        
        Proof: (related to Proposition: Addition of Real Numbers)
        
        Proposition: Addition Of Real Numbers Is Associative (1)
        
        Proof: (related to Proposition: Addition Of Real Numbers Is Associative)
        
        Proposition: Addition Of Real Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Addition Of Real Numbers Is Commutative)
        
        Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers) (1)
        
        Proof: (related to Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers))
        
        Proposition: Existence of Inverse Real Numbers With Respect to Addition (1)
        
        Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Addition)
        
        Proposition: Uniqueness of Real Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Real Zero)
        
        Proposition: Uniqueness of Negative Numbers (6)
        
        Proof: (related to Proposition: Uniqueness of Negative Numbers)
        
        Motivation: Motivation for the Proof of Uniqueness of Zero (related to Proposition: Uniqueness of Negative Numbers)
        
        Corollary: `$-0=0$` (related to Proposition: Uniqueness of Negative Numbers) (1)
        
        Proof: (related to Corollary: `$-0=0$`)
        
        Corollary: `$(-x)y=-(xy)$` (related to Proposition: Uniqueness of Negative Numbers) (2)
        
        Proof: (related to Corollary: `$(-x)y=-(xy)$`)
        
        Corollary: `$(-x)(-y)=xy$` (related to Corollary: `$(-x)y=-(xy)$`) (1)
        
        Proof: (related to Corollary: `$(-x)(-y)=xy$`)
        
        Corollary: `$-(-x)=x$` (related to Proposition: Uniqueness of Negative Numbers) (1)
        
        Proof: (related to Corollary: `$-(-x)=x$`)
        
        Proposition: Addition of Real Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Addition of Real Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law for Adding Real Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Real Numbers)
      - Definition: Subtraction of Real Numbers
      - Proposition: Multiplication of Real Numbers (14)
        
        Proof: (related to Proposition: Multiplication of Real Numbers)
        
        Proposition: Multiplication of Real Numbers Is Associative (2)
        
        Corollary: General Associative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Associative) (1)
        
        Proof: (related to Corollary: General Associative Law of Multiplication)
        
        Proof: (related to Proposition: Multiplication of Real Numbers Is Associative)
        
        Proposition: Multiplication of Real Numbers Is Commutative (2)
        
        Proof: (related to Proposition: Multiplication of Real Numbers Is Commutative)
        
        Corollary: General Commutative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Commutative) (1)
        
        Proof: (related to Corollary: General Commutative Law of Multiplication)
        
        Proposition: Multiplication of Real Numbers Is Cancellative (2)
        
        Proof: (related to Proposition: Multiplication of Real Numbers Is Cancellative)
        
        Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers (1)
        
        Proof: (related to Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers)
        
        Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers) (1)
        
        Proof: (related to Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers))
        
        Proposition: Uniqueness of Real One (1)
        
        Proof: (related to Proposition: Uniqueness of Real One)
        
        Proposition: Existence of Inverse Real Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Multiplication)
        
        Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication (3)
        
        Proof: (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
        
        Corollary: `$1^{-1}=1$` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication) (1)
        
        Proof: (related to Corollary: `$1^{-1}=1$`)
        
        Corollary: `$(x^{-1})^{-1}=x$` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication) (1)
        
        Proof: (related to Corollary: `$(x^{-1})^{-1}=x$`)
        
        Proposition: Multiplying Negative and Positive Real Numbers (1)
        
        Proof: (related to Proposition: Multiplying Negative and Positive Real Numbers)
      - Definition: Division of Real Numbers
      - Explanation: Why is it impossible to divide by `$0$`? (related to Proposition: Definition of Real Numbers)
      - Proposition: Distributivity Law For Real Numbers (3)
        
        Proof: (related to Proposition: Distributivity Law For Real Numbers)
        
        Corollary: `$0x=0$` (related to Proposition: Distributivity Law For Real Numbers) (2)
        
        Proof: (related to Corollary: `$0x=0$`)
        
        Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero. (related to Corollary: `$0x=0$`) (1)
        
        Proof: (related to Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero.)
      - Proposition: Unique Solvability of `$a+x=b$` (1)
        
        Proof: (related to Proposition: Unique Solvability of `$a+x=b$`)
      - Proposition: Unique Solvability of `$ax=b$` (1)
        
        Proof: (related to Proposition: Unique Solvability of `$ax=b$`)
      - Proposition: `$-(x+y)=-x-y$` (1)
        
        Proof: (related to Proposition: `$-(x+y)=-x-y$`)
      - Proposition: `$(xy)^{-1}=x^{-1}y^{-1}$` (1)
        
        Proof: (related to Proposition: `$(xy)^{-1}=x^{-1}y^{-1}$`)
    - Proposition: Algebraic Structure of Real Numbers Together with Addition (1)
      - Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition)
    - Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication (1)
      - Proof: (related to Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication)
    - Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication (2)
      - Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
      - Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
    - Definition: Order Relation of Real Numbers (2)
      - Corollary: Rules of Calculations with Inequalities (related to Definition: Order Relation of Real Numbers) (1)
        
        Proof: (related to Corollary: Rules of Calculations with Inequalities)
      - Proposition: Order Relation for Real Numbers is Strict and Total (1)
        
        Proof: (related to Proposition: Order Relation for Real Numbers is Strict and Total)
    - Axiom: Archimedean Axiom (5)
    - Definition: Absolute Value of Real Numbers (Modulus) (1)
      - Corollary: Properties of the Absolute Value (related to Definition: Absolute Value of Real Numbers (Modulus)) (1)
        
        Proof: (related to Corollary: Properties of the Absolute Value)
  - Part: Complex Numbers (29)
    - Chapter: Algebraic Properties of Complex Numbers (4)
    - Definition: Definition of Complex Numbers (11)
      - Definition: Addition of Complex Numbers (5)
        
        Proposition: Addition of Complex Numbers Is Associative (1)
        
        Proof: (related to Proposition: Addition of Complex Numbers Is Associative)
        
        Proposition: Addition of Complex Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Addition of Complex Numbers Is Commutative)
        
        Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers) (1)
        
        Proof: (related to Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers))
        
        Proposition: Existence of Inverse Complex Numbers With Respect to Addition (1)
        
        Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Addition)
        
        Proposition: Uniqueness of Complex Zero (1)
        
        Proof: (related to Proposition: Uniqueness of Complex Zero)
      - Definition: Subtraction of Complex Numbers
      - Definition: Multiplication of Complex Numbers (4)
        
        Proposition: Multiplication of Complex Numbers Is Associative (1)
        
        Proof: (related to Proposition: Multiplication of Complex Numbers Is Associative)
        
        Proposition: Multiplication of Complex Numbers Is Commutative (1)
        
        Proof: (related to Proposition: Multiplication of Complex Numbers Is Commutative)
        
        Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers) (1)
        
        Proof: (related to Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers))
        
        Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication (1)
        
        Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication)
      - Proposition: Distributivity Law for Complex Numbers (1)
        
        Proof: (related to Proposition: Distributivity Law for Complex Numbers)
    - Chapter: Simple Formulas Involving the Complex Numbers (5)
      - Proposition: Extracting the Real and the Imaginary Part of a Complex Number (1)
        
        Proof: (related to Proposition: Extracting the Real and the Imaginary Part of a Complex Number)
      - Proposition: Calculating with Complex Conjugates (1)
        
        Proof: (related to Proposition: Calculating with Complex Conjugates)
      - Proposition: Complex Numbers Cannot Be Ordered (1)
        
        Proof: (related to Proposition: Complex Numbers Cannot Be Ordered)
      - Proposition: Absolute Value of Complex Conjugate (1)
        
        Proof: (related to Proposition: Absolute Value of Complex Conjugate)
      - Proposition: Absolute Value of the Product of Complex Numbers (1)
        
        Proof: (related to Proposition: Absolute Value of the Product of Complex Numbers)
    - Chapter: Vector Properties of Complex Numbers (9)
      - Lemma: Linear Independence of the Imaginary Unit `$i$` and the Complex Number `$1$` (1)
        
        Proof: (related to Lemma: Linear Independence of the Imaginary Unit `$i$` and the Complex Number `$1$`)
      - Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `$1$` and Imaginary Unit `$i$` Form Their Basis (1)
        
        Proof: (related to Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `$1$` and Imaginary Unit `$i$` Form Their Basis)
      - Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers (1)
        
        Proof: (related to Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers)
      - Definition: Complex Conjugate
      - Definition: Dot Product of Complex Numbers
      - Definition: Absolute Value of Complex Numbers
      - Definition: Argument of a Complex Number
      - Proposition: Polar Coordinates of a Complex Number (1)
        
        Proof: (related to Proposition: Polar Coordinates of a Complex Number)
      - Proposition: Multiplication of Complex Numbers Using Polar Coordinates (1)
        
        Proof: (related to Proposition: Multiplication of Complex Numbers Using Polar Coordinates)
  - Explanation: Comparison Between the Number Systems (related to Branch: Number Systems and Arithmetics)
  - Part: Solving Strategies and Sample Solutions Related to Arithmetics (34)
    - Chapter: Product Manipulation Methods (1)
      - Definition: Products
    - Chapter: Some Important Constants (3)
      - Proposition: Imaginary Unit (1)
        
        Proof: (related to Proposition: Imaginary Unit)
      - Definition: Euler's Constant
      - Definition: Number `$\pi$`
    - Chapter: Sum Manipulation Methods (22)
      - Definition: Sums
      - Proposition: Basic Rules of Manipulating Finite Sums (1)
        
        Proof: (related to Proposition: Basic Rules of Manipulating Finite Sums)
      - Proposition: Double Summation (1)
        
        Proof: (related to Proposition: Double Summation)
      - Proposition: Rule of Combining Different Sets of Indices (1)
        
        Proof: (related to Proposition: Rule of Combining Different Sets of Indices)
      - Proposition: The General Perturbation Method (1)
        
        Proof: (related to Proposition: The General Perturbation Method)
      - Proposition: Abelian Partial Summation Method (1)
        
        Proof: (related to Proposition: Abelian Partial Summation Method)
      - Section: Closed Formulas for Sums (15)
        
        Proposition: Sum of Consecutive Natural Numbers (1)
        
        Proof: By Induction (related to Proposition: Sum of Consecutive Natural Numbers)
        
        Proposition: Sum of Consecutive Odd Numbers (1)
        
        Proof: By Induction (related to Proposition: Sum of Consecutive Odd Numbers)
        
        Proposition: Sum of Squares (1)
        
        Proof: (related to Proposition: Sum of Squares)
        
        Proposition: Sum of Cube Numbers (1)
        
        Proof: (related to Proposition: Sum of Cube Numbers)
        
        Proposition: Geometric Sum (1)
        
        Proof: By Induction (related to Proposition: Geometric Sum)
        
        Proposition: Sum of Geometric Progression (1)
        
        Proof: (related to Proposition: Sum of Geometric Progression)
        
        Proposition: Sum of Arithmetic Progression (1)
        
        Proof: (related to Proposition: Sum of Arithmetic Progression)
        
        Proposition: Alternating Sum of Binomial Coefficients (1)
        
        Proof: (related to Proposition: Alternating Sum of Binomial Coefficients)
        
        Proposition: Sum of Cosines (1)
        
        Proof: (related to Proposition: Sum of Cosines)
        
        Proposition: Sum of Binomial Coefficients (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients)
        
        Proposition: Sum of Binomial Coefficients I (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients I)
        
        Proposition: Sum of Binomial Coefficients II (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients II)
        
        Proposition: Sum of Binomial Coefficients III (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients III)
        
        Proposition: Sum of Binomial Coefficients IV (1)
        
        Proof: (related to Proposition: Sum of Binomial Coefficients IV)
        
        Proposition: Sum of Factorials (I) (1)
        
        Proof: By Induction (related to Proposition: Sum of Factorials (I))
      - Proposition: Product of Two Sums (Generalized Distributivity Rule) (1)
        
        Proof: (related to Proposition: Product of Two Sums (Generalized Distributivity Rule))
    - Chapter: Fractional Arithmetic (5)
      - Definition: Ratio of Two Real Numbers
      - Proposition: Equality of Two Ratios (1)
        
        Proof: (related to Proposition: Equality of Two Ratios)
      - Proposition: Sum and Difference of Two Ratios (1)
        
        Proof: (related to Proposition: Sum and Difference of Two Ratios)
      - Proposition: Product of Two Ratios (1)
        
        Proof: (related to Proposition: Product of Two Ratios)
      - Proposition: Ratio of Two Ratios (1)
        
        Proof: (related to Proposition: Ratio of Two Ratios)
    - Chapter: Sequences of Numbers (3)
      - Section: Fibonacci Numbers
      - Section: Fermat Numbers
      - Definition: Triangle Numbers
- Branch: Algebra (220)
  - Part: Algebraic Structures - Overview (54)
    - Motivation: Common Concepts Of Algebra: Substructures and Morphisms (related to Part: Algebraic Structures - Overview) (8)
      - Definition: Substructure (1)
        
        Proposition: Subset of Powers is a Submonoid (1)
        
        Proof: (related to Proposition: Subset of Powers is a Submonoid)
      - Explanation: All Types of Morphisms and Their Properties (related to Motivation: Common Concepts Of Algebra: Substructures and Morphisms)
      - Definition: Homomorphism
      - Definition: Monomorphism
      - Definition: Epimorphism
      - Definition: Isomorphism
      - Definition: Endomorphism
      - Definition: Automorphism
    - Definition: Algebraic Structure (Algebra)
    - Definition: Binary Operation
    - Chapter: Important Properties of Binary Operations (8)
      - Definition: Closure
      - Definition: Associativity (1)
        
        Corollary: General Associative Law (related to Definition: Associativity) (1)
        
        Proof: (related to Corollary: General Associative Law)
      - Definition: Commutativity (1)
        
        Corollary: General Commutative Law (related to Definition: Commutativity) (1)
        
        Proof: (related to Corollary: General Commutative Law)
      - Definition: Existence of a Neutral Element
      - Proposition: Uniqueness of the Neutral Element (1)
        
        Proof: Zero (Absorbing, Annihilating) Element, Left Zero, Right Zero (related to Proposition: Uniqueness of the Neutral Element)
      - Definition: Inverse Element
      - Proposition: Uniqueness of Inverse Elements (1)
        
        Proof: Conformity (related to Proposition: Uniqueness of Inverse Elements)
      - Definition: Cancellation Property
    - Explanation: Operation Table (related to Part: Algebraic Structures - Overview)
    - Chapter: Magmas, Semigroups, Monoids (Overview) (8)
      - Definition: Magma
      - Axiom: Axioms of Magma
      - Definition: Semigroup
      - Axiom: Axioms of Semigroup
      - Definition: Monoid
      - Axiom: Axioms of Monoid
      - Example: Examples of Magmas, Semigroups, and Monoids (related to Chapter: Magmas, Semigroups, Monoids (Overview))
      - Definition: Exponentiation in a Monoid
    - Chapter: Groups (Overview) (11)
      - Motivation: Calculations in a Group (related to Chapter: Groups (Overview)) (3)
        
        Proposition: Simple Calculations Rules in a Group (1)
        
        Proof: (related to Proposition: Simple Calculations Rules in a Group)
        
        Definition: Exponentiation in a Group (1)
        
        Corollary: Rules for Exponentiation in a Group (related to Definition: Exponentiation in a Group) (1)
        
        Proof: (related to Corollary: Rules for Exponentiation in a Group)
        
        Explanation: Calculation Rules in a Group with Additive Notation (related to Motivation: Calculations in a Group)
      - Example: Examples of Groups (related to Chapter: Groups (Overview))
      - Definition: Group
      - Axiom: Axioms of Group
      - Definition: Commutative (Abelian) Group
      - Definition: Subgroup
      - Proposition: Criteria for Subgroups (1)
        
        Proof: (related to Proposition: Criteria for Subgroups)
      - Explanation: Unions of Subgroups Are Not Subgroups (related to Chapter: Groups (Overview))
      - Proposition: Unique Solvability of `$a\ast x=b$` in Groups (1)
        
        Proof: (related to Proposition: Unique Solvability of `$a\ast x=b$` in Groups)
    - Chapter: Rings (Overview) (7)
      - Axiom: Axiom of Distributivity
      - Definition: (Unit) Ring
      - Definition: Commutative (Unit) Ring (1)
        
        Definition: Multiplicative System
      - Definition: Subring
      - Definition: Ring Homomorphism (2)
        
        Definition: Algebra over a Ring (2)
        
        Definition: Integral Element (1)
        
        Definition: Ring of Integers
        
        Definition: Integral Closure
      - Example: Examples of Ring Homomorphisms (related to Chapter: Rings (Overview))
    - Chapter: Fields (Overview) (5)
      - Example: Examples of Fields (related to Chapter: Fields (Overview))
      - Definition: Field
      - Definition: Subfield
      - Definition: Field Homomorphism
      - Proposition: In a Field, `$0$` Is Unequal `$1$` (1)
        
        Proof: (related to Proposition: In a Field, `$0$` Is Unequal `$1$`)
    - Chapter: Vector Spaces (Overview) (3)
      - Definition: Vector Space (2)
        
        Example: Examples of Vector Spaces (related to Definition: Vector Space)
        
        Proposition: Simple Consequences from the Definition of a Vector Space (1)
        
        Proof: (related to Proposition: Simple Consequences from the Definition of a Vector Space)
      - Definition: Subspace (1)
        
        Example: Trivial Subspaces, Zero Space (related to Definition: Subspace)
    - Chapter: Modules (Overview) (1)
      - Definition: Module
  - Part: Group Theory (32)
    - Definition: Generating Set of a Group
    - Definition: Group Homomorphism (6)
      - Example: Examples of Group Homomorphisms (related to Definition: Group Homomorphism)
      - Proposition: Properties of a Group Homomorphism (1)
        
        Proof: (related to Proposition: Properties of a Group Homomorphism)
      - Example: Examples of Properties of Group Homomorphisms (related to Definition: Group Homomorphism)
      - Lemma: Kernel and Image of Group Homomorphism (1)
        
        Proof: (related to Lemma: Kernel and Image of Group Homomorphism)
      - Example: Examples of Kernels and Images of Group Homomorphisms (related to Definition: Group Homomorphism)
      - Lemma: Kernel and Image of a Group Homomorphism are Subgroups (1)
        
        Proof: (related to Lemma: Kernel and Image of a Group Homomorphism are Subgroups)
    - Proposition: Additive Subgroups of Integers (1)
      - Proof: (related to Proposition: Additive Subgroups of Integers)
    - Theorem: Construction of Groups from Commutative and Cancellative Semigroups (1)
      - Proof: (related to Theorem: Construction of Groups from Commutative and Cancellative Semigroups)
    - Definition: Cyclic Group, Order of an Element
    - Example: Examples of Cyclic Groups (related to Part: Group Theory)
    - Proposition: Finite Order of an Element Equals Order Of Generated Group (1)
      - Proof: (related to Proposition: Finite Order of an Element Equals Order Of Generated Group)
    - Proposition: Group Homomorphisms with Cyclic Groups (1)
      - Proof: (related to Proposition: Group Homomorphisms with Cyclic Groups)
    - Lemma: Cyclic Groups are Abelian (1)
      - Proof: (related to Lemma: Cyclic Groups are Abelian)
    - Lemma: Subgroups of Cyclic Groups (1)
      - Proof: (related to Lemma: Subgroups of Cyclic Groups)
    - Proposition: Subgroups of Finite Cyclic Groups (1)
      - Proof: (related to Proposition: Subgroups of Finite Cyclic Groups)
    - Definition: Direct Product of Groups
    - Definition: Conjugate Elements of a Group
    - Definition: Cosets
    - Proposition: Properties of Cosets (1)
      - Proof: (related to Proposition: Properties of Cosets)
    - Lemma: Subgroups and Their Cosets are Equipotent (1)
      - Proof: (related to Lemma: Subgroups and Their Cosets are Equipotent)
    - Theorem: Order of Subgroup Divides Order of Finite Group (1)
      - Proof: (related to Theorem: Order of Subgroup Divides Order of Finite Group)
    - Theorem: Order of Cyclic Group (Fermat's Little Theorem) (1)
      - Proof: (related to Theorem: Order of Cyclic Group (Fermat's Little Theorem))
    - Chapter: Symmetry Groups
    - Definition: Normal Subgroups
    - Lemma: Factor Groups (1)
      - Proof: (related to Lemma: Factor Groups)
    - Lemma: Group Homomorphisms and Normal Subgroups (1)
      - Proof: (related to Lemma: Group Homomorphisms and Normal Subgroups)
    - Theorem: First Isomorphism Theorem for Groups (1)
      - Proof: (related to Theorem: First Isomorphism Theorem for Groups)
    - Theorem: Classification of Cyclic Groups (2)
      - Proof: (related to Theorem: Classification of Cyclic Groups)
      - Definition: Group Order
    - Theorem: Classification of Finite Groups with the Order of a Prime Number (1)
      - Proof: (related to Theorem: Classification of Finite Groups with the Order of a Prime Number)
    - Definition: Group Operation
  - Part: Algebraic Number Theory and Ring Theory (51)
    - Chapter: Divisibility in General Rings (34)
      - Definition: Zero Divisor and Integral Domain (4)
        
        Proposition: Cancellation Law (1)
        
        Proof: (related to Proposition: Cancellation Law)
        
        Definition: Zero Ring
        
        Theorem: Construction of Fields from Integral Domains (1)
        
        Proof: (related to Theorem: Construction of Fields from Integral Domains)
        
        Theorem: Finite Integral Domains are Fields (1)
        
        Proof: (related to Theorem: Finite Integral Domains are Fields)
      - Proposition: Generalization of Cancellative Multiplication of Integers (1)
        
        Proof: By Induction (related to Proposition: Generalization of Cancellative Multiplication of Integers)
      - Definition: Generalization of Divisor and Multiple
      - Definition: Generalization of the Greatest Common Divisor
      - Definition: Generalization of the Least Common Multiple
      - Definition: Unit (1)
        
        Proposition: Group of Units (1)
        
        Proof: (related to Proposition: Group of Units)
      - Definition: Associate (1)
        
        Lemma: A Criterion for Associates (1)
        
        Proof: By Induction (related to Lemma: A Criterion for Associates)
      - Definition: Irreducible, Prime
      - Definition: Factorial Ring, Generalization of Factorization
      - Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder
      - Section: Ideals (19)
        
        Definition: Ideal (3)
        
        Lemma: Greatest Common Divisor and Least Common Multiple of Ideals (1)
        
        Proof: (related to Lemma: Greatest Common Divisor and Least Common Multiple of Ideals)
        
        Definition: Divisibility of Ideals
        
        Definition: Addition of Ideals
        
        Proposition: Principal Ideals being Prime Ideals (1)
        
        Proof: By Induction (related to Proposition: Principal Ideals being Prime Ideals)
        
        Proposition: Principal Ideals being Maximal Ideals (1)
        
        Proof: By Induction (related to Proposition: Principal Ideals being Maximal Ideals)
        
        Definition: Principal Ideal (5)
        
        Proposition: Principal Ideal Generated by A Unit (1)
        
        Proof: By Induction (related to Proposition: Principal Ideal Generated by A Unit)
        
        Proposition: Criterions for Equality of Principal Ideals (1)
        
        Proof: By Induction (related to Proposition: Criterions for Equality of Principal Ideals)
        
        Lemma: Divisibility of Principal Ideals (1)
        
        Proof: (related to Lemma: Divisibility of Principal Ideals)
        
        Definition: Principal Ideal Ring
        
        Definition: Principal Ideal Domain
        
        Definition: Generating Set of an Ideal
        
        Definition: Prime Ideal (5)
        
        Definition: Spectrum of a Commutative Ring (3)
        
        Proposition: Spectrum Function of Commutative Rings (1)
        
        Proof: (related to Proposition: Spectrum Function of Commutative Rings)
        
        Proposition: Open and Closed Subsets of a Zariski Topology (1)
        
        Proof: (related to Proposition: Open and Closed Subsets of a Zariski Topology)
        
        Definition: Zariski Topology of a Commutative Ring
        
        Lemma: Fiber of Prime Ideals Under a Spectrum Function (2)
        
        Proof: (related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
        
        Lemma: Fiber of Prime Ideals (1)
        
        Proof: (related to Lemma: Fiber of Prime Ideals)
        
        Definition: Maximal Ideal (1)
        
        Lemma: Fiber of Maximal Ideals (1)
        
        Proof: (related to Lemma: Fiber of Maximal Ideals)
        
        Lemma: Prime Ideals of Multiplicative Systems in Integral Domains (1)
        
        Proof: (related to Lemma: Prime Ideals of Multiplicative Systems in Integral Domains)
        
        Theorem: Connection between Rings, Ideals, and Fields (1)
        
        Proof: (related to Theorem: Connection between Rings, Ideals, and Fields)
      - Lemma: Factor Rings, Generalization of Congruence Classes (2)
        
        Proof: (related to Lemma: Factor Rings, Generalization of Congruence Classes)
        
        Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring (1)
        
        Proof: (related to Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring)
    - Chapter: Polynomial Rings, Irreducibility, and Field Extensions (11)
      - Definition: Polynomial over a Ring, Degree, Variable (1)
        
        Definition: Reduction of an Integer Polynomial Modulo a Prime Number
      - Definition: Polynomial Ring
      - Definition: Elementary Symmetric Functions
      - Definition: Irreducible Polynomial
      - Definition: Field Extension (1)
        
        Definition: Finite Field Extension
      - Definition: Algebraic Element
      - Definition: Multiplicity of a Root of a Polynomial
      - Definition: Transcendental Element
      - Definition: Minimal Polynomial
      - Definition: Subadditive Function
      - Section: Solutions of Polynomials (1)
        
        Proposition: Quadratic Formula (1)
        
        Proof: (related to Proposition: Quadratic Formula)
    - Chapter: Algebraic and Transcendent Numbers (3)
      - Motivation: Rational Numbers and the Greatest Common Divisor (related to Chapter: Algebraic and Transcendent Numbers)
      - Definition: Continued Fractions
      - Lemma: Continuants and Convergents (1)
        
        Proof: By Induction (related to Lemma: Continuants and Convergents)
    - Theorem: Isomorphism of Rings
    - Definition: Characteristic of a Ring (2)
      - Explanation: Characteristic Zero Instead of Characteristic Infinite (related to Definition: Characteristic of a Ring)
      - Lemma: Any Positive Characteristic Is a Prime Number (1)
        
        Proof: (related to Lemma: Any Positive Characteristic Is a Prime Number)
  - Part: Finite Fields (3)
    - Definition: Finite Field
    - Definition: Characteristic of a Field
    - Definition: Prime Field
  - Part: Galois Theory
  - Part: Linear Algebra (65)
    - Chapter: Introduction to Matrices and Vectors (13)
      - Definition: Matrix, Set of Matrices over a Field
      - Definition: Transposed Matrix
      - Definition: Column Vectors and Row Vectors
      - Section: Addition of Matrices and Vectors (4)
        
        Definition: Matrix and Vector Addition
        
        Definition: Zero Matrix, Zero Vector
        
        Proposition: Abelian Group of Matrices Under Addition (2)
        
        Proof: (related to Proposition: Abelian Group of Matrices Under Addition)
        
        Corollary: Abelian Group of Vectors Under Addition (related to Proposition: Abelian Group of Matrices Under Addition) (1)
        
        Proof: (related to Corollary: Abelian Group of Vectors Under Addition)
      - Definition: Square Matrix (3)
        
        Definition: Symmetric Matrix
        
        Definition: Upper and Lower Triangular Matrix
        
        Definition: Diagonal Matrix
      - Definition: Matrix Multiplication
      - Definition: Identity Matrix
      - Definition: Invertible and Inverse Matrix
    - Chapter: Linear Equations and Systems of Linear Equations (SLEs) (14)
      - Definition: Linear Equations with many Unknowns (1)
        
        Corollary: Solutions of a Linear Equation with many Unknowns (related to Definition: Linear Equations with many Unknowns) (1)
        
        Proof: (related to Corollary: Solutions of a Linear Equation with many Unknowns)
      - Definition: Systems of Linear Equations with many Unknowns
      - Definition: Coefficient Matrix
      - Section: Solving Simple Systems of Linear Equations (5)
        
        Example: Solution to a Zero SLE (related to Section: Solving Simple Systems of Linear Equations)
        
        Example: Solution to a Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
        
        Example: Solution to a Degenerated Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
        
        Definition: Solution to an Upper Triangular SLE - Backward Substitution
        
        Definition: Solution to a Lower Triangular SLE - Forward Substitution
      - Section: Solving General Systems Of Linear Equations - Gaussian Method (4)
        
        Definition: Elementary Gaussian Operations
        
        Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE (1)
        
        Proof: (related to Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE)
        
        Definition: Gaussian Method to Solve Systems of Linear Equations, Rank of a Matrix
        
        Example: The Gaussian Method in Practice (related to Section: Solving General Systems Of Linear Equations - Gaussian Method)
      - Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs (1)
        
        Proof: (related to Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs)
      - Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations (1)
        
        Proof: By Induction (related to Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations)
    - Chapter: Vectors Revised - Vector Spaces (18)
      - Application: SLEs Revised in the Light of Vector Spaces (related to Chapter: Vectors Revised - Vector Spaces)
      - Definition: Linear Combination
      - Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector
      - Definition: Dimension of a Vector Space
      - Proposition: Quotient Space (1)
        
        Proof: (related to Proposition: Quotient Space)
      - Definition: Generating Systems
      - Definition: Basis, Coordinate System (2)
        
        Lemma: Uniqueness Lemma of a Finite Basis (1)
        
        Proof: (related to Lemma: Uniqueness Lemma of a Finite Basis)
        
        Theorem: Finite Basis Theorem (1)
        
        Proof: (related to Theorem: Finite Basis Theorem)
      - Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
      - Definition: Linear Span
      - Section: Cross Product
      - Section: Subspaces (2)
        
        Definition: Direct Sum of Vector Spaces
        
        Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace (1)
        
        Proof: (related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
      - Section: Dual Space
      - Definition: Linear Map (3)
        
        Explanation: Kernel and Image (related to Definition: Linear Map)
        
        Explanation: Matrices of Linear Maps (related to Definition: Linear Map)
        
        Definition: Multilinear Map (1)
        
        Definition: Alternating Multilinear Map
      - Definition: Bilinear Form (1)
        
        Definition: Symmetric Bilinear Form
    - Chapter: Vectors (2)
      - Definition: Dot Product, Inner Product, Scalar Product (General Field Case) (1)
        
        Corollary: Properties of a Real Scalar Product (related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case)) (1)
        
        Proof: (related to Corollary: Properties of a Real Scalar Product)
      - Definition: Dot Product, Inner Product, Scalar Product (Complex Case) (1)
        
        Proposition: Properties of a Complex Scalar Product (1)
        
        Proof: (related to Proposition: Properties of a Complex Scalar Product)
    - Chapter: Eigenvalues and Eigenvectors (2)
      - Definition: Eigenvalue
      - Definition: Eigenvector
    - Chapter: Principal Axis Transformation
    - Chapter: Jordan Normal Form
    - Chapter: Rotations and Basic Transformations
    - Chapter: Determinants (1)
      - Definition: Recursive Definition of the Determinant
    - Chapter: Affine Spaces (12)
      - Definition: Affine Subspace
      - Definition: Affine Basis, Affine Coordinate System (2)
        
        Corollary: Barycentric Coordinates, Barycenter (related to Definition: Affine Basis, Affine Coordinate System) (1)
        
        Proof: (related to Corollary: Barycentric Coordinates, Barycenter)
        
        Definition: Bounded Affine Set
      - Section: Affine Maps
      - Section: Normal Forms
      - Section: Quadrics
      - Definition: Affine Space (2)
        
        Definition: Euclidean Affine Space
        
        Definition: Unitary Affine Space
      - Definition: Affinely Dependent and Affinely Independent Points
      - Definition: Dimension of an Affine Space (1)
        
        Definition: Points, Lines, Planes, Hyperplanes
      - Definition: Convex Affine Set (2)
        
        Corollary: Intersection of Convex Affine Sets (related to Definition: Convex Affine Set) (1)
        
        Proof: (related to Corollary: Intersection of Convex Affine Sets)
        
        Definition: Convex Hull
  - Part: Constructions with Ruler and Compass
  - Part: Ordered Fields and Their Topology (11)
    - Definition: Ordered Field
    - Proposition: `$0$` Is Less Than `$1$` In Ordered Fields (1)
      - Proof: (related to Proposition: `$0$` Is Less Than `$1$` In Ordered Fields)
    - Proposition: Square of a Non-Zero Element is Positive in Ordered Fields (1)
      - Proof: (related to Proposition: Square of a Non-Zero Element is Positive in Ordered Fields)
    - Definition: Absolute Values and Non-Archimedean Absolute Values of Fields
    - Example: Examples of Absolute Values (related to Part: Ordered Fields and Their Topology)
    - Proposition: A Field with an Absolute Value is a Metric Space (1)
      - Proof: (related to Proposition: A Field with an Absolute Value is a Metric Space)
    - Definition: Dependent and Independent Absolute Values
    - Proposition: Characterization of Dependent Absolute Values (1)
      - Proof: (related to Proposition: Characterization of Dependent Absolute Values)
    - Proposition: Characterization of Non-Archimedean Absolute Values (1)
      - Proof: (related to Proposition: Characterization of Non-Archimedean Absolute Values)
    - Definition: Signum Function in An Ordered Field
    - Definition: Complete Ordered Field
  - Part: Solving Strategies and Sample Solutions to Problems in Algebra (2)
    - Chapter: Group-theoretic Problems (2)
      - Problem: Verifying Group Properties (1)
        
        Solution: (related to Problem: Verifying Group Properties)
      - Problem: Verifying Subgroup Properties (1)
        
        Solution: (related to Problem: Verifying Subgroup Properties)
- Branch: Analysis (450)
  - Part: Historical Development of Analysis (3)
  - Part: Real Analysis of One Variable and Elements of Complex Analysis (311)
    - Chapter: Basics of Real Analysis of One Variable (14)
      - Section: Real Intervals and Bounded Real Sets (11)
        
        Definition: Real Intervals
        
        Definition: Supremum, Least Upper Bound
        
        Definition: Maximum (Real Numbers)
        
        Definition: Extended Real Numbers
        
        Definition: Supremum of Extended Real Numbers
        
        Definition: Infimum, Greatest Lower Bound
        
        Definition: Minimum (Real Numbers)
        
        Definition: Infimum of Extended Real Numbers
        
        Proposition: Closed Formula for the Maximum and Minimum of Two Numbers (1)
        
        Proof: (related to Proposition: Closed Formula for the Maximum and Minimum of Two Numbers)
        
        Definition: Nested Real Intervals
        
        Proposition: Limit of Nested Real Intervals (1)
        
        Proof: (related to Proposition: Limit of Nested Real Intervals)
      - Proposition: Rational Numbers are Dense in Real Numbers (1)
        
        Proof: (related to Proposition: Rational Numbers are Dense in Real Numbers)
      - Proposition: Open Intervals Contain Uncountably Many Irrational Numbers (1)
        
        Proof: (related to Proposition: Open Intervals Contain Uncountably Many Irrational Numbers)
      - Proposition: Open Real Intervals are Uncountable (1)
        
        Proof: (related to Proposition: Open Real Intervals are Uncountable)
    - Chapter: Real-valued Sequences and Limits of Sequences and Functions (54)
      - Definition: Real Sequence
      - Definition: Convergent Real Sequence
      - Definition: Divergent Sequences (5)
        
        Example: Divergent Alternating Sequence (related to Definition: Divergent Sequences)
        
        Definition: Sequences Tending To Infinity
        
        Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero (1)
        
        Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero)
        
        Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty (1)
        
        Proof: (related to Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty)
        
        Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity (1)
        
        Proof: (related to Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity)
      - Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
      - Definition: Monotonic Sequences (1)
        
        Proposition: Real Sequences Contain Monotonic Subsequences (1)
        
        Proof: (related to Proposition: Real Sequences Contain Monotonic Subsequences)
      - Section: Theorems Regarding Limits Of Sequences (8)
        
        Proposition: Uniqueness Of the Limit of a Sequence (1)
        
        Proof: (related to Proposition: Uniqueness Of the Limit of a Sequence)
        
        Proposition: Sum of Convergent Real Sequences (1)
        
        Proof: (related to Proposition: Sum of Convergent Real Sequences)
        
        Proposition: Difference of Convergent Real Sequences (1)
        
        Proof: (related to Proposition: Difference of Convergent Real Sequences)
        
        Proposition: Product of Convegent Real Sequences (1)
        
        Proof: (related to Proposition: Product of Convegent Real Sequences)
        
        Proposition: Product of a Real Number and a Convergent Real Sequence (1)
        
        Proof: (related to Proposition: Product of a Real Number and a Convergent Real Sequence)
        
        Proposition: Quotient of Convergent Real Sequences (1)
        
        Proof: (related to Proposition: Quotient of Convergent Real Sequences)
        
        Proposition: How Convergence Preserves the Order Relation of Sequence Members (1)
        
        Proof: (related to Proposition: How Convergence Preserves the Order Relation of Sequence Members)
        
        Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members (1)
        
        Proof: (related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)
      - Definition: Limits of Real Functions
      - Section: Theorems Regarding Limits of Functions (8)
        
        Proposition: Limit of a Function is Unique If It Exists (1)
        
        Proof: (related to Proposition: Limit of a Function is Unique If It Exists)
        
        Proposition: Arithmetic of Functions with Limits - Sums (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Sums)
        
        Proposition: Arithmetic of Functions with Limits - Difference (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Difference)
        
        Proposition: Arithmetic of Functions with Limits - Division (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Division)
        
        Proposition: Preservation of Inequalities for Limits of Functions (1)
        
        Proof: (related to Proposition: Preservation of Inequalities for Limits of Functions)
        
        Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity (1)
        
        Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity)
        
        Theorem: Squeezing Theorem for Functions (1)
        
        Proof: (related to Theorem: Squeezing Theorem for Functions)
        
        Proposition: Arithmetic of Functions with Limits - Product (1)
        
        Proof: (related to Proposition: Arithmetic of Functions with Limits - Product)
      - Section: Examples of Limit Calculations (17)
        
        Proposition: Limit of 1/n (1)
        
        Proof: (related to Proposition: Limit of 1/n)
        
        Proposition: Limit of the Constant Function (1)
        
        Proof: (related to Proposition: Limit of the Constant Function)
        
        Proposition: Limit of the Identity Function (1)
        
        Proof: (related to Proposition: Limit of the Identity Function)
        
        Proposition: Limit of Nth Powers (1)
        
        Proof: By Induction (related to Proposition: Limit of Nth Powers)
        
        Proposition: Convergence Behavior of the Sequence `$(b^n)$` (1)
        
        Proof: (related to Proposition: Convergence Behavior of the Sequence `$(b^n)$`)
        
        Proposition: Limit of a Polynomial (1)
        
        Proof: By Induction (related to Proposition: Limit of a Polynomial)
        
        Proposition: Limit of a Rational Function (1)
        
        Proof: (related to Proposition: Limit of a Rational Function)
        
        Proposition: Limit of Nth Root of a Positive Constant (1)
        
        Proof: (related to Proposition: Limit of Nth Root of a Positive Constant)
        
        Proposition: Limit of Nth Root of N (1)
        
        Proof: (related to Proposition: Limit of Nth Root of N)
        
        Proposition: Limits of General Powers (2)
        
        Proof: (related to Proposition: Limits of General Powers)
        
        Corollary: Value of Zero to the Power of X (related to Proposition: Limits of General Powers) (1)
        
        Proof: (related to Corollary: Value of Zero to the Power of X)
        
        Proposition: Square Roots (1)
        
        Proof: (related to Proposition: Square Roots)
        
        Proposition: Limit of Exponential Growth as Compared to Polynomial Growth (1)
        
        Proof: (related to Proposition: Limit of Exponential Growth as Compared to Polynomial Growth)
        
        Proposition: Limits of Logarithm in `$[0,+\infty]$` (1)
        
        Proof: (related to Proposition: Limits of Logarithm in `$[0,+\infty]$`)
        
        Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth (1)
        
        Proof: (related to Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth)
        
        Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function (1)
        
        Proof: (related to Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function)
        
        Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function (1)
        
        Proof: (related to Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function)
      - Definition: Real Subsequence
      - Theorem: Every Bounded Real Sequence has a Convergent Subsequence (1)
        
        Proof: By Induction (related to Theorem: Every Bounded Real Sequence has a Convergent Subsequence)
      - Definition: Accumulation Point (Real Numbers) (4)
        
        Example: Examples of Accumulation Points (related to Definition: Accumulation Point (Real Numbers))
        
        Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence (1)
        
        Proof: (related to Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence)
        
        Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence (1)
        
        Proof: (related to Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence)
        
        Definition: Isolated Point (Real Numbers)
      - Definition: Asymptotical Approximation
      - Lemma: Decreasing Sequence of Suprema of Extended Real Numbers (1)
        
        Proof: (related to Lemma: Decreasing Sequence of Suprema of Extended Real Numbers)
      - Definition: Limit Superior
      - Lemma: Increasing Sequence of Infima of Extended Real Numbers (1)
        
        Proof: (related to Lemma: Increasing Sequence of Infima of Extended Real Numbers)
      - Definition: Limit Inferior
    - Chapter: Completeness of Real Numbers (7)
      - Proposition: The distance of real numbers makes real numbers a metric space. (1)
        
        Proof: (related to Proposition: The distance of real numbers makes real numbers a metric space.)
      - Definition: Real Cauchy Sequence
      - Theorem: Completeness Principle for Real Numbers (2)
        
        Proof: (related to Theorem: Completeness Principle for Real Numbers)
        
        Corollary: Non-Cauchy Sequences are Not Convergent (related to Theorem: Completeness Principle for Real Numbers) (1)
        
        Proof: (related to Corollary: Non-Cauchy Sequences are Not Convergent)
      - Theorem: Supremum Property, Infimum Property (1)
        
        Proof: (related to Theorem: Supremum Property, Infimum Property)
      - Proposition: Convergent Real Sequences Are Cauchy Sequences (1)
        
        Proof: (related to Proposition: Convergent Real Sequences Are Cauchy Sequences)
      - Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers (1)
        
        Proof: (related to Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers)
    - Chapter: Criteria for Convergence of Sequences (4)
      - Theorem: Every Bounded Monotonic Sequence Is Convergent (2)
        
        Proof: (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
        
        Application: Application of Monotonic Convergence (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
      - Proposition: Convergent Real Sequences are Bounded (1)
        
        Proof: (related to Proposition: Convergent Real Sequences are Bounded)
      - Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence (1)
        
        Proof: (related to Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence)
    - Theorem: Defining Properties of the Field of Real Numbers (1)
      - Proof: (related to Theorem: Defining Properties of the Field of Real Numbers)
    - Chapter: Useful Inequalities (21)
      - Proposition: Generalized Triangle Inequality (1)
        
        Proof: By Induction (related to Proposition: Generalized Triangle Inequality)
      - Theorem: Triangle Inequality (1)
        
        Proof: (related to Theorem: Triangle Inequality)
      - Theorem: Reverse Triangle Inequalities (1)
        
        Proof: (related to Theorem: Reverse Triangle Inequalities)
      - Definition: (Weighted) Arithmetic Mean
      - Theorem: Inequality of the Arithmetic Mean (1)
        
        Proof: (related to Theorem: Inequality of the Arithmetic Mean)
      - Theorem: Inequality of Weighted Arithmetic Mean (1)
        
        Proof: (related to Theorem: Inequality of Weighted Arithmetic Mean)
      - Theorem: Bernoulli's Inequality (1)
        
        Proof: By Induction (related to Theorem: Bernoulli's Inequality)
      - Proposition: Generalized Bernoulli's Inequality (1)
        
        Proof: By Induction (related to Proposition: Generalized Bernoulli's Inequality)
      - Proposition: Cauchy–Schwarz Inequality (1)
        
        Proof: (related to Proposition: Cauchy–Schwarz Inequality)
      - Definition: Geometric Mean
      - Theorem: Inequality Between the Geometric and the Arithmetic Mean (1)
        
        Proof: By Induction (related to Theorem: Inequality Between the Geometric and the Arithmetic Mean)
      - Lemma: Upper Bound for the Product of General Powers (1)
        
        Proof: (related to Lemma: Upper Bound for the Product of General Powers)
      - Proposition: Hölder's Inequality (1)
        
        Proof: (related to Proposition: Hölder's Inequality)
      - Proposition: Minkowski's Inequality (1)
        
        Proof: (related to Proposition: Minkowski's Inequality)
      - Proposition: Hölder's Inequality for Integral p-norms (1)
        
        Proof: (related to Proposition: Hölder's Inequality for Integral p-norms)
      - Proposition: Cauchy-Schwarz Inequality for Integral p-norms (1)
        
        Proof: (related to Proposition: Cauchy-Schwarz Inequality for Integral p-norms)
      - Proposition: Minkowski's Inequality for Integral p-norms (1)
        
        Proof: (related to Proposition: Minkowski's Inequality for Integral p-norms)
      - Proposition: Inequality between Square Numbers and Powers of `$2$` (1)
        
        Proof: By Induction (related to Proposition: Inequality between Square Numbers and Powers of `$2$`)
      - Proposition: Inequality between Powers of `$2$` and Factorials (1)
        
        Proof: By Induction (related to Proposition: Inequality between Powers of `$2$` and Factorials)
      - Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials (1)
        
        Proof: (related to Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials)
      - Proposition: Bounds for Partial Sums of Exponential Series (1)
        
        Proof: (related to Proposition: Bounds for Partial Sums of Exponential Series)
    - Chapter: Properties of Real Functions (77)
      - Definition: Continuous Functions at Single Real Numbers (9)
        
        Example: Examples of Functions Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
        
        Example: Examples of Functions Not Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
        
        Proposition: Compositions of Continuous Functions on a Whole Domain (1)
        
        Proof: (related to Proposition: Compositions of Continuous Functions on a Whole Domain)
        
        Proposition: Composition of Continuous Functions at a Single Point (1)
        
        Proof: (related to Proposition: Composition of Continuous Functions at a Single Point)
        
        Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point (1)
        
        Proof: (related to Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point)
        
        Theorem: Intermediate Value Theorem (1)
        
        Proof: (related to Theorem: Intermediate Value Theorem)
        
        Theorem: Intermediate Root Value Theorem (1)
        
        Proof: By Induction (related to Theorem: Intermediate Root Value Theorem)
        
        Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point (related to Definition: Continuous Functions at Single Real Numbers) (1)
        
        Proof: (related to Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point)
        
        Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments (related to Definition: Continuous Functions at Single Real Numbers) (1)
        
        Proof: (related to Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments)
      - Definition: Continuous Real Functions (4)
        
        Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions (1)
        
        Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions)
        
        Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain (1)
        
        Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain)
        
        Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals (1)
        
        Proof: (related to Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals)
        
        Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth (1)
        
        Proof: (related to Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth)
      - Definition: Even and Odd Complex Functions
      - Section: Differentiable Functions (15)
        
        Definition: Difference Quotient
        
        Definition: Derivative, Differentiable Functions (1)
        
        Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability
        
        Definition: Higher-Order Derivatives (1)
        
        Proposition: Generalized Product Rule (1)
        
        Proof: (related to Proposition: Generalized Product Rule)
        
        Definition: Local Extremum (3)
        
        Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum (2)
        
        Proof: (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
        
        Explanation: Local Extrema on a Closed Interval (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
        
        Proposition: Sufficient Condition for a Local Extremum (1)
        
        Proof: (related to Proposition: Sufficient Condition for a Local Extremum)
        
        Theorem: Rolle's Theorem (1)
        
        Proof: (related to Theorem: Rolle's Theorem)
        
        Proposition: Differentiable Functions and Tangent-Linear Approximation (2)
        
        Proof: (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
        
        Corollary: Differentiable Functions are Continuous (related to Proposition: Differentiable Functions and Tangent-Linear Approximation) (1)
        
        Proof: (related to Corollary: Differentiable Functions are Continuous)
        
        Proposition: Characterization of Monotonic Functions via Derivatives (1)
        
        Proof: (related to Proposition: Characterization of Monotonic Functions via Derivatives)
        
        Theorem: Darboux's Theorem (3)
        
        Proof: (related to Theorem: Darboux's Theorem)
        
        Corollary: Estimating the Growth of a Function with its Derivative (related to Theorem: Darboux's Theorem) (2)
        
        Proof: (related to Corollary: Estimating the Growth of a Function with its Derivative)
        
        Corollary: Sufficient Condition for a Function to be Constant (related to Corollary: Estimating the Growth of a Function with its Derivative) (1)
        
        Proof: (related to Corollary: Sufficient Condition for a Function to be Constant)
        
        Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule (1)
        
        Proof: (related to Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule)
        
        Proposition: Chain Rule (1)
        
        Proof: (related to Proposition: Chain Rule)
      - Definition: Continuously Differentiable Functions
      - Definition: Uniformly Continuous Functions (Real Case) (5)
        
        Proposition: Not all Continuous Functions are also Uniformly Continuous (1)
        
        Proof: (related to Proposition: Not all Continuous Functions are also Uniformly Continuous)
        
        Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions (1)
        
        Proof: (related to Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions)
        
        Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous (2)
        
        Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
        
        Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
        
        Corollary: All Uniformly Continuous Functions are Continuous (related to Definition: Uniformly Continuous Functions (Real Case)) (1)
        
        Proof: (related to Corollary: All Uniformly Continuous Functions are Continuous)
      - Definition: Periodic Functions
      - Definition: Even and Odd Functions (1)
        
        Proposition: Derivatives of Even and Odd Functions (1)
        
        Proof: (related to Proposition: Derivatives of Even and Odd Functions)
      - Section: Integrable Functions (24)
        
        Subsection: Riemann Integral (22)
        
        Proposition: Riemann Integral for Step Functions (2)
        
        Proof: (related to Proposition: Riemann Integral for Step Functions)
        
        Proposition: Linearity and Monotony of the Riemann Integral for Step Functions (1)
        
        Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral for Step Functions)
        
        Definition: Riemann-Integrable Functions (8)
        
        Proposition: Step Function on Closed Intervals are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Step Function on Closed Intervals are Riemann-Integrable)
        
        Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable)
        
        Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable)
        
        Proposition: Linearity and Monotony of the Riemann Integral (1)
        
        Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral)
        
        Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable (1)
        
        Proof: (related to Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable)
        
        Proposition: Product of Riemann-integrable Functions is Riemann-integrable (1)
        
        Proof: (related to Proposition: Product of Riemann-integrable Functions is Riemann-integrable)
        
        Example: Existence of not Riemann-Integrable Functions (related to Definition: Riemann-Integrable Functions)
        
        Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions (1)
        
        Proof: (related to Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions)
        
        Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions (2)
        
        Proof: (related to Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions)
        
        Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals (1)
        
        Proof: (related to Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals)
        
        Definition: Riemann Sum With Respect to a Partition (1)
        
        Proposition: Riemann Sum Converging To the Riemann Integral (1)
        
        Proof: (related to Proposition: Riemann Sum Converging To the Riemann Integral)
        
        Theorem: Indefinite Integral, Antiderivative (2)
        
        Proof: (related to Theorem: Indefinite Integral, Antiderivative)
        
        Proposition: Antiderivatives are Uniquely Defined Up to a Constant (1)
        
        Proof: By Induction (related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
        
        Theorem: Fundamental Theorem of Calculus (1)
        
        Proof: (related to Theorem: Fundamental Theorem of Calculus)
        
        Proposition: Integrals on Adjacent Intervals (1)
        
        Proof: By Induction (related to Proposition: Integrals on Adjacent Intervals)
        
        Theorem: Integration by Substitution (1)
        
        Proof: (related to Theorem: Integration by Substitution)
        
        Theorem: Mean Value Theorem For Riemann Integrals (1)
        
        Proof: (related to Theorem: Mean Value Theorem For Riemann Integrals)
        
        Theorem: Partial Integration (1)
        
        Proof: (related to Theorem: Partial Integration)
        
        Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine (1)
        
        Proof: (related to Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine)
        
        Lemma: Trapezoid Rule (1)
        
        Proof: (related to Lemma: Trapezoid Rule)
        
        Subsection: Extended Concept of the Riemann Integral - the Improper Integral (1)
        
        Definition: Improper Integral
        
        Subsection: Riemann-Stieltjes Integral
      - Lemma: Invertible Functions on Real Intervals (2)
        
        Proof: (related to Lemma: Invertible Functions on Real Intervals)
        
        Proposition: Derivative of an Invertible Function on Real Invervals (1)
        
        Proof: (related to Proposition: Derivative of an Invertible Function on Real Invervals)
      - Definition: Convex and Concave Functions (2)
        
        Proposition: Convexity and Concaveness Test (1)
        
        Proof: (related to Proposition: Convexity and Concaveness Test)
        
        Proposition: Convex Functions on Open Intervals are Continuous (1)
        
        Proof: (related to Proposition: Convex Functions on Open Intervals are Continuous)
      - Definition: Logarithmically Convex and Concave Functions
      - Section: Uniform Convergence of Functions (8)
        
        Definition: Pointwise and Uniformly Convergent Sequences of Functions
        
        Example: Pointwise vs. Uniformly Convergent Sequences of Functions (related to Section: Uniform Convergence of Functions)
        
        Proposition: Only the Uniform Convergence Preserves Continuity (1)
        
        Proof: (related to Proposition: Only the Uniform Convergence Preserves Continuity)
        
        Definition: Supremum Norm for Functions
        
        Proposition: Supremum Norm and Uniform Convergence (1)
        
        Proof: (related to Proposition: Supremum Norm and Uniform Convergence)
        
        Proposition: Uniform Convergence Criterion of Cauchy (1)
        
        Proof: (related to Proposition: Uniform Convergence Criterion of Cauchy)
        
        Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series (1)
        
        Proof: (related to Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series)
        
        Proposition: Calculations with Uniformly Convergent Functions (1)
        
        Proof: (related to Proposition: Calculations with Uniformly Convergent Functions)
      - Definition: Monotonic Functions
      - Definition: Bounded and Unbounded Functions (2)
        
        Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals (2)
        
        Proof: (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
        
        Corollary: Continuous Real Functions on Closed Intervals are Bounded (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals) (1)
        
        Proof: (related to Corollary: Continuous Real Functions on Closed Intervals are Bounded)
    - Chapter: Types of Real Functions (86)
      - Definition: Constant Function Real Case (1)
        
        Corollary: Derivative of a Constant Function (related to Definition: Constant Function Real Case) (1)
        
        Proof: (related to Corollary: Derivative of a Constant Function)
      - Definition: Real Identity Function (1)
        
        Proposition: Identity Function is Continuous (1)
        
        Proof: (related to Proposition: Identity Function is Continuous)
      - Definition: Reciprocal Function (2)
        
        Proposition: Derivative of the Reciprocal Function (1)
        
        Proof: (related to Proposition: Derivative of the Reciprocal Function)
        
        Proposition: Integral of the Reciprocal Function (1)
        
        Proof: (related to Proposition: Integral of the Reciprocal Function)
      - Definition: Real Absolute Value Function (1)
        
        Proposition: Derivate of Absolute Value Function Does Not Exist at `$0$` (1)
        
        Proof: (related to Proposition: Derivate of Absolute Value Function Does Not Exist at `$0$`)
      - Definition: Step Functions (2)
        
        Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval (1)
        
        Proof: (related to Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval)
        
        Definition: Order Relation for Step Functions
      - Definition: Positive and Negative Parts of a Real-Valued Function
      - Definition: Linear Function (2)
        
        Corollary: Derivative of a Linear Function `$ax+b$` (related to Definition: Linear Function) (2)
        
        Proof: (related to Corollary: Derivative of a Linear Function `$ax+b$`)
        
        Proof: (related to Corollary: Derivative of a Linear Function `$ax+b$`)
      - Proposition: Nth Roots of Positive Numbers (1)
        
        Proof: (related to Proposition: Nth Roots of Positive Numbers)
      - Proposition: Rational Powers of Positive Numbers (2)
        
        Proof: (related to Proposition: Rational Powers of Positive Numbers)
        
        Corollary: Limit of N-th Roots (related to Proposition: Rational Powers of Positive Numbers) (1)
        
        Proof: (related to Corollary: Limit of N-th Roots)
      - Proposition: Nth Powers (2)
        
        Proof: (related to Proposition: Nth Powers)
        
        Proposition: Derivative of the n-th Power Function (1)
        
        Proof: (related to Proposition: Derivative of the n-th Power Function)
      - Proposition: General Powers of Positive Numbers (4)
        
        Proof: (related to Proposition: General Powers of Positive Numbers)
        
        Proposition: Calculation Rules for General Powers (1)
        
        Proof: (related to Proposition: Calculation Rules for General Powers)
        
        Proposition: Derivative of General Powers of Positive Numbers (1)
        
        Proof: (related to Proposition: Derivative of General Powers of Positive Numbers)
        
        Proposition: Integral of General Powers (1)
        
        Proof: (related to Proposition: Integral of General Powers)
      - Definition: Polynomials (3)
        
        Proposition: Limits of Polynomials at Infinity (2)
        
        Proof: (related to Proposition: Limits of Polynomials at Infinity)
        
        Corollary: Real Polynomials of Odd Degree Have at Least One Real Root (related to Proposition: Limits of Polynomials at Infinity) (1)
        
        Proof: (related to Corollary: Real Polynomials of Odd Degree Have at Least One Real Root)
        
        Proposition: Eveness (Oddness) of Polynomials (1)
        
        Proof: (related to Proposition: Eveness (Oddness) of Polynomials)
      - Definition: Rational Functions (1)
        
        Proposition: Rational Functions are Continuous (1)
        
        Proof: (related to Proposition: Rational Functions are Continuous)
      - Proposition: Exponential Function (18)
        
        Proof: (related to Proposition: Exponential Function)
        
        Definition: Exponential Function of General Base (6)
        
        Proposition: Continuity of Exponential Function of General Base (1)
        
        Proof: (related to Proposition: Continuity of Exponential Function of General Base)
        
        Proposition: Functional Equation of the Exponential Function of General Base (2)
        
        Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base)
        
        Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function of General Base) (1)
        
        Proof: (related to Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property)
        
        Proposition: Exponential Function of General Base With Natural Exponents (1)
        
        Proof: By Induction (related to Proposition: Exponential Function of General Base With Natural Exponents)
        
        Proposition: Exponential Function of General Base With Integer Exponents (1)
        
        Proof: (related to Proposition: Exponential Function of General Base With Integer Exponents)
        
        Proposition: Functional Equation of the Exponential Function of General Base (Revised) (1)
        
        Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base (Revised))
        
        Proposition: Estimate for the Remainder Term of Exponential Function (1)
        
        Proof: (related to Proposition: Estimate for the Remainder Term of Exponential Function)
        
        Proposition: Functional Equation of the Exponential Function (6)
        
        Proof: (related to Proposition: Functional Equation of the Exponential Function)
        
        Corollary: Reciprocity of Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function) (2)
        
        Proof: (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
        
        Corollary: (Real) Exponential Function Is Always Positive (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property) (1)
        
        Proof: (related to Corollary: (Real) Exponential Function Is Always Positive)
        
        Corollary: Exponential Function Is Strictly Monotonically Increasing (related to Proposition: Functional Equation of the Exponential Function) (1)
        
        Proof: (related to Corollary: Exponential Function Is Strictly Monotonically Increasing)
        
        Corollary: Exponential Function Is Non-Negative (Real Case) (related to Proposition: Functional Equation of the Exponential Function) (1)
        
        Proof: (related to Corollary: Exponential Function Is Non-Negative (Real Case))
        
        Corollary: Exponential Function and the Euler Constant (related to Proposition: Functional Equation of the Exponential Function) (1)
        
        Proof: (related to Corollary: Exponential Function and the Euler Constant)
        
        Proposition: Continuity of Exponential Function (1)
        
        Proof: (related to Proposition: Continuity of Exponential Function)
        
        Proposition: `$\exp(0)=1$` (1)
        
        Proof: (related to Proposition: `$\exp(0)=1$`)
        
        Proposition: Derivative of the Exponential Function (1)
        
        Proof: (related to Proposition: Derivative of the Exponential Function)
        
        Proposition: Integral of the Exponential Function (1)
        
        Proof: (related to Proposition: Integral of the Exponential Function)
      - Section: Logarithms (5)
        
        Proposition: Natural Logarithm (4)
        
        Proof: (related to Proposition: Natural Logarithm)
        
        Proposition: Functional Equation of the Natural Logarithm (1)
        
        Proof: (related to Proposition: Functional Equation of the Natural Logarithm)
        
        Proposition: Derivative of the Natural Logarithm (1)
        
        Proof: (related to Proposition: Derivative of the Natural Logarithm)
        
        Proposition: Integral of the Natural Logarithm (1)
        
        Proof: (related to Proposition: Integral of the Natural Logarithm)
        
        Proposition: Logarithm to a General Base (1)
        
        Proof: (related to Proposition: Logarithm to a General Base)
      - Section: Trigonometric Functions (30)
        
        Definition: Cosine of a Real Variable (5)
        
        Proposition: Eveness of the Cosine of a Real Variable (1)
        
        Proof: (related to Proposition: Eveness of the Cosine of a Real Variable)
        
        Proposition: Zero of Cosine (1)
        
        Proof: (related to Proposition: Zero of Cosine)
        
        Proposition: Derivative of Cosine (1)
        
        Proof: (related to Proposition: Derivative of Cosine)
        
        Proposition: Integral of Cosine (1)
        
        Proof: (related to Proposition: Integral of Cosine)
        
        Corollary: Representing Real Cosine by Complex Exponential Function (related to Definition: Cosine of a Real Variable) (1)
        
        Proof: (related to Corollary: Representing Real Cosine by Complex Exponential Function)
        
        Definition: Sine of a Real Variable (4)
        
        Proposition: Oddness of the Sine of a Real Variable (1)
        
        Proof: (related to Proposition: Oddness of the Sine of a Real Variable)
        
        Proposition: Derivative of Sine (1)
        
        Proof: (related to Proposition: Derivative of Sine)
        
        Proposition: Integral of Sine (1)
        
        Proof: (related to Proposition: Integral of Sine)
        
        Corollary: Representing Real Sine by Complex Exponential Function (related to Definition: Sine of a Real Variable) (1)
        
        Proof: (related to Corollary: Representing Real Sine by Complex Exponential Function)
        
        Definition: Tangent of a Real Variable (2)
        
        Proposition: Additivity Theorem of Tangent (1)
        
        Proof: (related to Proposition: Additivity Theorem of Tangent)
        
        Proposition: Derivative of Tangent (1)
        
        Proof: (related to Proposition: Derivative of Tangent)
        
        Proposition: Continuity of Cosine and Sine (1)
        
        Proof: (related to Proposition: Continuity of Cosine and Sine)
        
        Proposition: Pythagorean Identity (1)
        
        Proof: (related to Proposition: Pythagorean Identity)
        
        Proposition: Additivity Theorems of Cosine and Sine (1)
        
        Proof: (related to Proposition: Additivity Theorems of Cosine and Sine)
        
        Proposition: Infinite Series for Cosine and Sine (2)
        
        Proof: (related to Proposition: Infinite Series for Cosine and Sine)
        
        Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine (1)
        
        Proof: (related to Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine)
        
        Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function (6)
        
        Proof: (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
        
        Corollary: Cosine and Sine are Periodic Functions (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Cosine and Sine are Periodic Functions)
        
        Corollary: Negative Cosine and Sine vs Shifting the Argument (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Negative Cosine and Sine vs Shifting the Argument)
        
        Corollary: Arguments for which Cosine and Sine are Equal to Each Other (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Arguments for which Cosine and Sine are Equal to Each Other)
        
        Corollary: All Zeros of Cosine and Sine (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: All Zeros of Cosine and Sine)
        
        Corollary: More Insight to Euler's Identity (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
        
        Proof: (related to Corollary: More Insight to Euler's Identity)
        
        Proposition: Inverse Cosine of a Real Variable (1)
        
        Proof: (related to Proposition: Inverse Cosine of a Real Variable)
        
        Proposition: Inverse Sine of a Real Variable (3)
        
        Proof: (related to Proposition: Inverse Sine of a Real Variable)
        
        Proposition: Derivative of the Inverse Sine (1)
        
        Proof: (related to Proposition: Derivative of the Inverse Sine)
        
        Proposition: Integral of Inverse Sine (1)
        
        Proof: (related to Proposition: Integral of Inverse Sine)
        
        Proposition: Inverse Tangent of a Real Variable (4)
        
        Proof: (related to Proposition: Inverse Tangent of a Real Variable)
        
        Proposition: Inverse Tangent and Complex Exponential Function (1)
        
        Proof: (related to Proposition: Inverse Tangent and Complex Exponential Function)
        
        Proposition: Derivative of the Inverse Tangent (1)
        
        Proof: (related to Proposition: Derivative of the Inverse Tangent)
        
        Proposition: Integral of the Inverse Tangent (1)
        
        Proof: (related to Proposition: Integral of the Inverse Tangent)
      - Section: Cyclometric Functions
      - Section: Hyporbolic Functions (3)
        
        Definition: Hyperbolic Cosine (1)
        
        Proposition: Sum of Arguments of Hyperbolic Cosine (1)
        
        Proof: (related to Proposition: Sum of Arguments of Hyperbolic Cosine)
        
        Definition: Hyperbolic Sine (1)
        
        Proposition: Sum of Arguments of Hyperbolic Sine (1)
        
        Proof: (related to Proposition: Sum of Arguments of Hyperbolic Sine)
        
        Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine (1)
        
        Proof: (related to Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine)
      - Section: Inverse Hyperbolic Functions (2)
        
        Proposition: Inverse Hyperbolic Sine (1)
        
        Proof: (related to Proposition: Inverse Hyperbolic Sine)
        
        Proposition: Inverse Hyperbolic Cosine (1)
        
        Proof: (related to Proposition: Inverse Hyperbolic Cosine)
      - Section: Mixed Functions
      - Proposition: Gamma Function (2)
        
        Proposition: Gamma Function Interpolates the Factorial (1)
        
        Proof: (related to Proposition: Gamma Function Interpolates the Factorial)
        
        Proof: (related to Proposition: Gamma Function)
      - Example: Examples of Real Functions, Whose Graphs Cannot be Plotted (related to Chapter: Types of Real Functions)
    - Chapter: Infinite Series - Overview (40)
      - Definition: Convergent Complex Series (1)
        
        Proposition: A General Criterion for the Convergence of Infinite Complex Series (1)
        
        Proof: (related to Proposition: A General Criterion for the Convergence of Infinite Complex Series)
      - Definition: Infinite Series, Partial Sums
      - Definition: Convergent Real Series (4)
        
        Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent (1)
        
        Proof: (related to Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent)
        
        Proposition: Sum of Convergent Real Series (1)
        
        Proof: (related to Proposition: Sum of Convergent Real Series)
        
        Proposition: Difference of Convergent Real Series (1)
        
        Proof: (related to Proposition: Difference of Convergent Real Series)
        
        Proposition: Product of a Real Number and a Convergent Real Series (1)
        
        Proof: (related to Proposition: Product of a Real Number and a Convergent Real Series)
      - Definition: Absolutely Convergent Series (2)
        
        Proposition: Convergence Behaviour of Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Convergence Behaviour of Absolutely Convergent Series)
        
        Proposition: Cauchy Product of Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Series)
      - Definition: Complex Infinite Series
      - Definition: Absolutely Convergent Complex Series (3)
        
        Proposition: Direct Comparison Test For Absolutely Convergent Complex Series (1)
        
        Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Complex Series)
        
        Proposition: Ratio Test For Absolutely Convergent Complex Series (1)
        
        Proof: By Induction (related to Proposition: Ratio Test For Absolutely Convergent Complex Series)
        
        Proposition: Cauchy Product of Absolutely Convergent Complex Series (1)
        
        Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Complex Series)
      - Definition: Divergent Series
      - Section: Convergence and Divergence Criteria for Real Series (20)
        
        Proposition: Monotony Criterion (2)
        
        Corollary: Monotony Criterion for Absolute Series (related to Proposition: Monotony Criterion) (1)
        
        Proof: (related to Corollary: Monotony Criterion for Absolute Series)
        
        Proof: (related to Proposition: Monotony Criterion)
        
        Proposition: Leibniz Criterion for Alternating Series (1)
        
        Proof: (related to Proposition: Leibniz Criterion for Alternating Series)
        
        Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero (1)
        
        Proof: (related to Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero)
        
        Proposition: Cauchy Condensation Criterion (2)
        
        Example: Applications of the Cauchy Condensation Criterion (related to Proposition: Cauchy Condensation Criterion)
        
        Proof: (related to Proposition: Cauchy Condensation Criterion)
        
        Proposition: Direct Comparison Test For Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Series)
        
        Proposition: Direct Comparison Test For Divergent Series (1)
        
        Proof: (related to Proposition: Direct Comparison Test For Divergent Series)
        
        Proposition: Limit Comparizon Test (1)
        
        Proof: (related to Proposition: Limit Comparizon Test)
        
        Proposition: Root Test (1)
        
        Proof: (related to Proposition: Root Test)
        
        Proposition: Ratio Test (1)
        
        Proof: By Induction (related to Proposition: Ratio Test)
        
        Proposition: Limit Test for Roots or Ratios (1)
        
        Proof: (related to Proposition: Limit Test for Roots or Ratios)
        
        Lemma: Convergence Test for Telescoping Series (1)
        
        Proof: (related to Lemma: Convergence Test for Telescoping Series)
        
        Proposition: Raabe's Test (1)
        
        Proof: (related to Proposition: Raabe's Test)
        
        Subsection: Convergence Criteria for Infinite Series Involving Products (4)
        
        Lemma: Abel's Lemma for Testing Convergence (1)
        
        Proof: (related to Lemma: Abel's Lemma for Testing Convergence)
        
        Proposition: Cauchy-Schwarz Test (1)
        
        Proof: (related to Proposition: Cauchy-Schwarz Test)
        
        Proposition: Abel's Test (1)
        
        Proof: (related to Proposition: Abel's Test)
        
        Proposition: Dirichlet's Test (1)
        
        Proof: (related to Proposition: Dirichlet's Test)
        
        Proposition: Integral Test for Convergence (1)
        
        Proof: (related to Proposition: Integral Test for Convergence)
        
        Proposition: Cauchy Criterion (1)
        
        Proof: (related to Proposition: Cauchy Criterion)
      - Definition: Rearrangement of Infinite Series (2)
        
        Proposition: Rearrangement of Absolutely Convergent Series (1)
        
        Proof: (related to Proposition: Rearrangement of Absolutely Convergent Series)
        
        Proposition: Rearrangement of Convergent Series (1)
        
        Proof: (related to Proposition: Rearrangement of Convergent Series)
      - Section: Closed Formulas for Infinite Real Series (1)
        
        Proposition: Infinite Geometric Series (1)
        
        Proof: (related to Proposition: Infinite Geometric Series)
      - Definition: `$b$`-Adic Fractions (4)
        
        Proposition: `$b$`-Adic Fractions Are Real Cauchy Sequences (1)
        
        Proof: (related to Proposition: `$b$`-Adic Fractions Are Real Cauchy Sequences)
        
        Proposition: Unique Representation of Real Numbers as `$b$`-adic Fractions (2)
        
        Proof: (related to Proposition: Unique Representation of Real Numbers as `$b$`-adic Fractions)
        
        Corollary: Real Numbers Can Be Approximated by Rational Numbers (related to Proposition: Unique Representation of Real Numbers as `$b$`-adic Fractions) (1)
        
        Proof: (related to Corollary: Real Numbers Can Be Approximated by Rational Numbers)
        
        Definition: Decimal Representation of Real Numbers
    - Chapter: Infinite Products - Overview
    - Chapter: Representation of Functions as Taylor Series (4)
      - Proposition: Approximation of Functions by Taylor's Formula (1)
        
        Proof: (related to Proposition: Approximation of Functions by Taylor's Formula)
      - Proposition: Taylor's Formula with Remainder Term of Lagrange (1)
        
        Proof: (related to Proposition: Taylor's Formula with Remainder Term of Lagrange)
      - Theorem: Taylor's Formula (2)
        
        Corollary: Taylor's Formula for Polynomials (related to Theorem: Taylor's Formula) (1)
        
        Proof: (related to Corollary: Taylor's Formula for Polynomials)
        
        Proof: By Induction (related to Theorem: Taylor's Formula)
    - Chapter: Power Series Introduction
    - Chapter: Fourier Series
  - Part: Real Analysis of Multiple Variables (17)
    - Chapter: Differentiability (6)
      - Section: Implicit Functions
      - Definition: Directional Derivative (1)
        
        Definition: Higher Order Directional Derivative
      - Section: Extremal Values with Side Conditions
      - Definition: `$n$` times Continuously Differentiable Functions
      - Definition: Totally Differentiable Functions, Total Derivative
      - Definition: Derivative of an n-Dimensional Curve
    - Chapter: Integrability (4)
      - Section: Path Integrals
      - Section: Multiple Riemann Integrals
      - Section: Multiple Lebesgue Integrals
      - Section: Jordan Measurability
    - Chapter: Fixed Point Theory (3)
      - Section: Brouwer's Theorem
      - Section: Schauder's Theorem
      - Section: Kakutani's Theorem
    - Definition: Curves In the Multidimensional Space `$\mathbb R^n$` (2)
      - Definition: Jordan Arc (Simple Curve)
      - Definition: Closed Curve, Open Curve
    - Definition: Generalized Polynomial Function
    - Proposition: Definition of the Metric Space `$\mathbb R^n$`, Euclidean Norm (1)
      - Proof: (related to Proposition: Definition of the Metric Space `$\mathbb R^n$`, Euclidean Norm)
  - Part: Complex Analysis (68)
    - Chapter: Topological Aspects (25)
      - Definition: Open and Closed Discs
      - Definition: Closed and Open Regions of the Complex Plane
      - Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
      - Proposition: The distance of complex numbers makes complex numbers a metric space. (1)
        
        Proof: (related to Proposition: The distance of complex numbers makes complex numbers a metric space.)
      - Proposition: Complex Convergent Sequences are Bounded (1)
        
        Proof: (related to Proposition: Complex Convergent Sequences are Bounded)
      - Definition: Continuous Functions at Single Complex Numbers
      - Proposition: Convergent Complex Sequences Are Cauchy Sequences (1)
        
        Proof: (related to Proposition: Convergent Complex Sequences Are Cauchy Sequences)
      - Definition: Complex Sequence
      - Definition: Bounded Complex Sequences (1)
        
        Proposition: Convergent Complex Sequences Are Bounded (1)
        
        Proof: (related to Proposition: Convergent Complex Sequences Are Bounded)
      - Definition: Convergent Complex Sequence (5)
        
        Proposition: Quotient of Convergent Complex Sequences (1)
        
        Proof: (related to Proposition: Quotient of Convergent Complex Sequences)
        
        Proposition: Sum of Convergent Complex Sequences (1)
        
        Proof: (related to Proposition: Sum of Convergent Complex Sequences)
        
        Proposition: Difference of Convergent Complex Sequences (1)
        
        Proof: (related to Proposition: Difference of Convergent Complex Sequences)
        
        Proposition: Product of Convegent Complex Sequences (1)
        
        Proof: (related to Proposition: Product of Convegent Complex Sequences)
        
        Proposition: Product of a Complex Number and a Convergent Complex Sequence (1)
        
        Proof: (related to Proposition: Product of a Complex Number and a Convergent Complex Sequence)
      - Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences (2)
        
        Proof: (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
        
        Corollary: Convergence of Complex Conjugate Sequence (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences) (1)
        
        Proof: (related to Corollary: Convergence of Complex Conjugate Sequence)
      - Definition: Limits of Complex Functions
      - Definition: Complex Cauchy Sequence
      - Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences (1)
        
        Proof: (related to Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences)
      - Theorem: Completeness Principle for Complex Numbers (1)
        
        Proof: (related to Theorem: Completeness Principle for Complex Numbers)
      - Definition: Accumulation Points (Complex Numbers)
      - Section: Paths
      - Section: Homology and Winding Numbers
      - Section: Homotopy
      - Definition: Bounded Complex Sets
    - Chapter: Geometric Aspects (6)
      - Section: Moebius Transformations (3)
        
        Subsection: Inversion
        
        Subsection: Fixed Points
        
        Subsection: Cross-Ratio
      - Section: Riemann's Plane and Sphere
      - Section: Isogonality
      - Section: Analytic Continuation
    - Chapter: Properties of Complex Functions (9)
      - Definition: Continuous Complex Functions
      - Section: Differentiable Complex Functions
      - Section: Periodic Complex Functions
      - Section: Invertible Complex Functions
      - Section: Integrable Complex Functions (2)
        
        Subsection: Indefinite Integral
        
        Subsection: Path Integrals
      - Section: Holomorphic Functions (3)
        
        Subsection: Criteria for Holomorphic Functions
        
        Subsection: Properties
        
        Subsection: Harmonic Functions
    - Chapter: Types of Complex Functions (20)
      - Proposition: Complex Exponential Function (7)
        
        Proof: (related to Proposition: Complex Exponential Function)
        
        Proposition: Estimate for the Remainder Term of Complex Exponential Function (1)
        
        Proof: (related to Proposition: Estimate for the Remainder Term of Complex Exponential Function)
        
        Proposition: Functional Equation of the Complex Exponential Function (2)
        
        Proof: (related to Proposition: Functional Equation of the Complex Exponential Function)
        
        Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Complex Exponential Function) (1)
        
        Proof: (related to Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property)
        
        Proposition: `$\exp(0)=1$` (Complex Case) (1)
        
        Proof: (related to Proposition: `$\exp(0)=1$` (Complex Case))
        
        Proposition: Continuity of Complex Exponential Function (1)
        
        Proof: (related to Proposition: Continuity of Complex Exponential Function)
        
        Proposition: Complex Conjugate of Complex Exponential Function (1)
        
        Proof: (related to Proposition: Complex Conjugate of Complex Exponential Function)
      - Lemma: Unit Circle (1)
        
        Proof: (related to Lemma: Unit Circle)
      - Proposition: Euler's Formula (1)
        
        Proof: (related to Proposition: Euler's Formula)
      - Proposition: n-th Roots of Unity (1)
        
        Proof: (related to Proposition: n-th Roots of Unity)
      - Theorem: De Moivre's Identity, Complex Powers (1)
        
        Proof: By Induction (related to Theorem: De Moivre's Identity, Complex Powers)
      - Section: Complex Trigonometric Functions
      - Definition: Complex Polynomials
      - Section: Complex Rational Functions
      - Section: Complex Logarithms
      - Section: Complex Cyclometric Functions
      - Section: Complex Hyporbolic Functions
      - Section: Complex Inverse Hyperbolic Functions
      - Section: Complex Mixed Functions
      - Lemma: Euler's Identity (1)
        
        Proof: (related to Lemma: Euler's Identity)
    - Chapter: Laurant Series
    - Chapter: Isolated Singularities
    - Chapter: Meromorphic Functions
    - Chapter: Residue Theorem
    - Chapter: Discrete Fourier Transform (DFT) (4)
      - Definition: Even Complex Sequence
      - Definition: Odd Complex Sequence
      - Definition: n-Periodical Complex Sequence
      - Lemma: Sum of Roots Of Unity in Complete Residue Systems (1)
        
        Proof: (related to Lemma: Sum of Roots Of Unity in Complete Residue Systems)
  - Part: Differential Equations (14)
    - Chapter: Classification of Differential Equations (5)
      - Definition: First-Order Ordinary Differential Equation (ODE) (1)
        
        Proposition: Differential Equation of the Exponential Function (1)
        
        Proof: (related to Proposition: Differential Equation of the Exponential Function)
      - Section: Linear vs. Non-Linear DE
      - Section: Order of DE
      - Section: Systems of DE
      - Proposition: Legendre Polynomials and Legendre Differential Equations (1)
        
        Proof: (related to Proposition: Legendre Polynomials and Legendre Differential Equations)
    - Chapter: Methods to Solve Ordinary DE (7)
      - Section: Linear 1. Order DE
      - Section: Linear 2. Order DE
      - Section: Linear n-th Order DE
      - Section: Linear Systems of 1. Order DE
      - Section: Non-Linear DE
      - Section: Boundary Value Problems
      - Definition: Solution of Ordinary DE
    - Chapter: Methods to Solve Partial DE (2)
      - Section: Linear 1. Order DE
      - Section: Linear 2. Order DE
  - Part: Linear Integral Equations (4)
    - Chapter: Fredholm Integral Equations
    - Chapter: Volterra Integral Equation
    - Chapter: Liouville-Neumann Series
    - Chapter: Abel Integral Equations
  - Part: Functional Analysis (29)
    - Theorem: Nested Closed Subset Theorem (1)
      - Proof: (related to Theorem: Nested Closed Subset Theorem)
    - Chapter: Banach Spaces and Banach Algebra (2)
      - Definition: Complete Metric Space
      - Definition: Banach Space
    - Chapter: Hilbert Spaces (1)
      - Definition: Hilbert Space
    - Chapter: Linear Operators and Linear Functionals
    - Chapter: Non-linear Operator
    - Chapter: Lebesgue Measure and Lebesgue Integral
    - Chapter: Simple Connectivity
    - Chapter: The `$C(X)$` Algebra
    - Chapter: Compact Sets (13)
      - Definition: Heine-Borel Property Defines Compact Subsets (1)
        
        Explanation: Explanation of the Heine-Borel Property (related to Definition: Heine-Borel Property Defines Compact Subsets)
      - Proposition: Image of a Compact Set Under a Continuous Function (1)
        
        Proof: (related to Proposition: Image of a Compact Set Under a Continuous Function)
      - Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space (1)
        
        Proof: (related to Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space)
      - Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space (1)
        
        Proof: (related to Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space)
      - Proposition: Closed n-Dimensional Cuboids Are Compact (2)
        
        Proof: (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
        
        Corollary: Closed Real Intervals Are Compact (related to Proposition: Closed n-Dimensional Cuboids Are Compact) (1)
        
        Proof: (related to Corollary: Closed Real Intervals Are Compact)
      - Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed (1)
        
        Proof: (related to Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed)
      - Proposition: Convergent Sequences are Bounded (1)
        
        Proof: (related to Proposition: Convergent Sequences are Bounded)
      - Theorem: Heine-Borel Theorem (1)
        
        Proof: (related to Theorem: Heine-Borel Theorem)
      - Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum (1)
        
        Proof: (related to Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum)
      - Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains (2)
        
        Proof: (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
        
        Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains) (1)
        
        Proof: (related to Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded)
      - Proposition: Closed Subsets of Compact Sets are Compact (1)
        
        Proof: (related to Proposition: Closed Subsets of Compact Sets are Compact)
    - Definition: Sigma-Algebra (5)
      - Definition: Measurable Set
      - Definition: Measureable Function
      - Definition: Measure (1)
        
        Definition: Finite and Sigma-Finite Measure
      - Definition: Pre-measure (1)
        
        Definition: Finite and Sigma-Finite Pre-measure
      - Definition: Measurable Space
    - Definition: Ring of Sets (measure-theoretic definition)
    - Definition: Functional (1)
      - Definition: Functional Equation
  - Part: Calculus of Variations
  - Part: Vector Analysis (3)
    - Definition: Vector Field
    - Chapter: Differential Operations on Vector Fields
    - Chapter: Integral Operations on Vector Fields
- Branch: Topology (86)
  - Part: Historical Development of Topology
  - Part: The Basics Set-theoretic Topology (63)
    - Chapter: Basic Topological Concepts (13)
      - Definition: Topological Space, Topology
      - Proposition: Clopen Sets and Boundaries (1)
        
        Proof: (related to Proposition: Clopen Sets and Boundaries)
      - Definition: Discrete and Indiscrete Topology
      - Definition: Ordering of Topologies
      - Definition: Open, Closed, Clopen
      - Proposition: Alternative Characterization of Topological Spaces (1)
        
        Proof: (related to Proposition: Alternative Characterization of Topological Spaces)
      - Definition: Neighborhood
      - Proposition: Properties of the Set of All Neighborhoods of a Point (1)
        
        Proof: (related to Proposition: Properties of the Set of All Neighborhoods of a Point)
      - Proposition: A Necessary Condition of a Neighborhood to be Open (1)
        
        Proof: (related to Proposition: A Necessary Condition of a Neighborhood to be Open)
      - Definition: Boundary Points, Closures, Interiors, and Exteriors
      - Proposition: How the Boundary Changes the Property of a Set of Being Open (1)
        
        Proof: (related to Proposition: How the Boundary Changes the Property of a Set of Being Open)
      - Definition: Regular Open, Regular Closed
      - Explanation: Fourteen Sets Formed By Closure, Interior and Complement Operations (related to Chapter: Basic Topological Concepts)
    - Chapter: Construction of Topologies (6)
      - Definition: Topological Subspaces and Subspace Topologies
      - Definition: Hereditary and Weakly Hereditary Properties
      - Proposition: Construction of Topological Spaces Using a Subbasis (1)
        
        Proof: (related to Proposition: Construction of Topological Spaces Using a Subbasis)
      - Definition: Topological Sum, Disjoint Union
      - Definition: Topological Product, Product Topology
      - Definition: Subbasis and Basis of Topology
    - Chapter: Sequences and Limits (12)
      - Definition: Isolated, Adherent, Limit, `$\omega$`-Accumulation and Condensation Points
      - Definition: Sequence
      - Definition: Carrier Set
      - Definition: Subsequence
      - Definition: Limits and Accumulation Points of Sequences
      - Example: Examples of Convergent Sequences in Topological Spaces (related to Chapter: Sequences and Limits)
      - Definition: Derived, Dense-in-itself, and Perfect Sets
      - Proposition: Perfect Sets vs. Derived Sets (1)
        
        Proof: (related to Proposition: Perfect Sets vs. Derived Sets)
      - Definition: Comparison of Filters, Finer and Coarser Filters
      - Axiom: Filter
      - Proposition: Filter Base (1)
        
        Proof: (related to Proposition: Filter Base)
      - Definition: Ultrafilter
    - Chapter: Density and Countability (2)
      - Definition: Dense Sets, Nowhere Dense Sets
      - Definition: First and Second Category Sets
    - Chapter: Continuity (9)
      - Definition: Continuous Function
      - Example: Example of Continuous Functions in Topological Spaces (related to Chapter: Continuity)
      - Proposition: Equivalent Notions of Continuous Functions (1)
        
        Proof: (related to Proposition: Equivalent Notions of Continuous Functions)
      - Proposition: Continuity of Compositions of Functions (1)
        
        Proof: (related to Proposition: Continuity of Compositions of Functions)
      - Definition: Open and Closed Functions
      - Proposition: Bijective Open Functions (1)
        
        Proof: (related to Proposition: Bijective Open Functions)
      - Definition: Homeomorphism, Homeomorphic Spaces
      - Proposition: Equivalent Notions of Homeomorphisms (1)
        
        Proof: (related to Proposition: Equivalent Notions of Homeomorphisms)
      - Definition: Topological, Continuous, Open, and Closed Invariants
    - Definition: Limit of a Function
    - Definition: Convergent Sequences and Limits (1)
      - Lemma: Criteria for Convergent Sequences (1)
        
        Proof: (related to Lemma: Criteria for Convergent Sequences)
    - Proposition: Uniqueness of the Limit of a Sequence (1)
      - Proof: (related to Proposition: Uniqueness of the Limit of a Sequence)
    - Definition: Cauchy Sequence (1)
      - Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces) (1)
        
        Proof: (related to Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces))
    - Theorem: Theorem of Bolzano-Weierstrass (1)
      - Proof: (related to Theorem: Theorem of Bolzano-Weierstrass)
    - Definition: Norm, Normed Vector Space (3)
      - Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm (2)
        
        Proof: (related to Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm)
        
        Proposition: Maximum Norm as a Limit of p-Norms (1)
        
        Proof: (related to Proposition: Maximum Norm as a Limit of p-Norms)
      - Proposition: Integral p-Norm (1)
        
        Proof: (related to Proposition: Integral p-Norm)
    - Definition: Isometry (1)
      - Proposition: Isometry is Injective (1)
        
        Proof: (related to Proposition: Isometry is Injective)
    - Definition: Bounded Sequence
    - Definition: Bounded Subset of a Metric Space
    - Proposition: Metric Spaces are Hausdorff Spaces (1)
      - Proof: (related to Proposition: Metric Spaces are Hausdorff Spaces)
    - Proposition: Distance in Normed Vector Spaces (1)
      - Proof: (related to Proposition: Distance in Normed Vector Spaces)
    - Definition: Open Cover
    - Definition: Continuous Functions in Metric Spaces (6)
      - Definition: Pointwise and Uniform Convergence (2)
        
        Example: A pointwise convergent sequence of functions, which is not uniformly convergent (related to Definition: Pointwise and Uniform Convergence)
        
        Corollary: Every uniformly convergent sequence of functions is pointwise convergent. (related to Definition: Pointwise and Uniform Convergence) (1)
        
        Proof: (related to Corollary: Every uniformly convergent sequence of functions is pointwise convergent.)
      - Proposition: $\epsilon$-$\delta$ Definition of Continuity (4)
        
        Proof: (related to Proposition: $\epsilon$-$\delta$ Definition of Continuity)
        
        Definition: Modulus of Continuity of a Continuous Function (3)
        
        Proposition: Modulus of Continuity is Subadditive (1)
        
        Proof: (related to Proposition: Modulus of Continuity is Subadditive)
        
        Proposition: Modulus of Continuity is Monotonically Increasing (1)
        
        Proof: (related to Proposition: Modulus of Continuity is Monotonically Increasing)
        
        Proposition: Modulus of Continuity is Continuous (1)
        
        Proof: (related to Proposition: Modulus of Continuity is Continuous)
    - Definition: Uniformly Continuous Functions (General Metric Spaces Case) (1)
      - Theorem: Continuous Functions on Compact Domains are Uniformly Continuous (1)
        
        Proof: (related to Theorem: Continuous Functions on Compact Domains are Uniformly Continuous)
  - Part: Separation Of Topological Spaces (5)
    - Axiom: Separation Axioms
    - Proposition: Characterization of `$T_1$` Spaces (1)
      - Proof: (related to Proposition: Characterization of `$T_1$` Spaces)
    - Proposition: Inheritance of the `$T_1$` Property (1)
      - Proof: (related to Proposition: Inheritance of the `$T_1$` Property)
    - Proposition: Characterization of `$T_2$` Spaces (1)
      - Proof: (related to Proposition: Characterization of `$T_2$` Spaces)
    - Proposition: Inheritance of the `$T_2$` Property (1)
      - Proof: (related to Proposition: Inheritance of the `$T_2$` Property)
  - Part: Metric Spaces (7)
    - Definition: Metric (Distance) (1)
      - Corollary: Every Distance Is Positive Definite (related to Definition: Metric (Distance)) (1)
        
        Proof: (related to Corollary: Every Distance Is Positive Definite)
    - Definition: Metric Space
    - Definition: Open Sets in Metric Spaces (2)
      - Lemma: Characterization of Closed Sets by Limits of Sequences (1)
        
        Proof: (related to Lemma: Characterization of Closed Sets by Limits of Sequences)
      - Definition: Open Function, Closed Function
    - Definition: Open Ball, Neighborhood
    - Definition: Diameter In Metric Spaces
    - Proposition: Metric Spaces and Empty Sets are Clopen (1)
      - Proof: (related to Proposition: Metric Spaces and Empty Sets are Clopen)
  - Part: Homotopy (9)
    - Definition: Manifold (6)
      - Definition: Differentiable Manifold, Atlas
      - Definition: `$C^n$` Differentiable Function
      - Definition: Tangent Bundle
      - Definition: Cotangent Bundle
      - Definition: Section over a Base Space
      - Definition: Differential Form of Degree k
    - Definition: Topological Chart
    - Definition: Transition Map
    - Definition: `$C^{n}$`-Diffeomorphism
  - Part: Algebraic Topology (1)
    - Definition: Simplex
- Branch: Geometry (700)
  - Part: Historical Development of Geometry
  - Part: Euclidean Geometry (682)
    - Chapter: Euclid's “Elements” (670)
      - Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines (106)
        
        Subsection: Definitions from Book 1 (28)
        
        Definition: 1.01: Point (1)
        
        Explanation: How a point is different from a solid, a surface and a line? (related to Definition: 1.01: Point)
        
        Definition: 1.02: Line, Curve (1)
        
        Explanation: How a line is different from a solid and a surface? (related to Definition: 1.02: Line, Curve)
        
        Definition: 1.03: Intersections of Lines
        
        Definition: 1.04: Straight Line, Segment and Ray (1)
        
        Definition: Collinear Points, Segments, Rays
        
        Definition: 1.05: Surface
        
        Definition: 1.06: Intersections of Surfaces
        
        Definition: 1.07: Plane
        
        Definition: 1.08: Plane Angle
        
        Definition: 1.09: Angle, Rectilinear, Vertex, Legs (3)
        
        Definition: Exterior, Interior, Alternate and Corresponding Angles
        
        Definition: Sum of Angles
        
        Definition: Supplemental Angles
        
        Definition: 1.10: Right Angle, Perpendicular Straight Lines (1)
        
        Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines) (1)
        
        Proof: By Euclid (related to Corollary: The supplemental angle of a right angle is another right angle.)
        
        Definition: 1.11: Obtuse Angle
        
        Definition: 1.12: Acute Angle
        
        Definition: 1.13: Boundary
        
        Definition: 1.14: Plane Figure
        
        Definition: 1.15: Circle, Circumference, Radius
        
        Definition: 1.16: Center of the Circle (1)
        
        Definition: Concentric Circles
        
        Definition: 1.17: Diameter of the Circle
        
        Definition: 1.18: Semicircle
        
        Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure (4)
        
        Definition: Triangle
        
        Definition: Pentagon
        
        Definition: Hexagon
        
        Definition: Decagon
        
        Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle (1)
        
        Definition: Altitude of a Triangle
        
        Definition: 1.21: Right Triangle, Obtuse Triangle, Acute Triangle
        
        Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium (1)
        
        Definition: Diagonal
        
        Definition: 1.23: Parallel Straight Lines
        
        Subsection: Axioms from Book 1 (5)
        
        Axiom: 1.1: Straight Line Determined by Two Distinct Points (1)
        
        Explanation: Why did Euclid postulate the axiom of straight line determined by two points? (related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
        
        Axiom: 1.2: Segment Extension
        
        Axiom: 1.3: Circle Determined by its Center and its Radius
        
        Axiom: 1.4: Equality of all Right Angles
        
        Axiom: 1.5: Parallel Postulate
        
        Subsection: Common Notions (all Books) (5)
        
        Explanation: 1.1: Equality is an Equivalence Relation (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.2: Adding Equations Preserves Equality (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.3: Subtracting Equations Preserves Equality (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.4: Congruent Figures (related to Subsection: Common Notions (all Books))
        
        Explanation: 1.5: Comparing the Size of Sets and Their Subsets (related to Subsection: Common Notions (all Books))
        
        Subsection: Propositions from Book 1 (68)
        
        Proposition: 1.01: Constructing an Equilateral Triangle (1)
        
        Proof: By Euclid (related to Proposition: 1.01: Constructing an Equilateral Triangle)
        
        Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment (1)
        
        Proof: By Euclid (related to Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment)
        
        Proposition: 1.03: Cutting a Segment at a Given Size (1)
        
        Proof: By Euclid (related to Proposition: 1.03: Cutting a Segment at a Given Size)
        
        Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle (1)
        
        Proof: By Euclid (related to Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle)
        
        Proposition: 1.05: Isosceles Triangles I (2)
        
        Proof: By Euclid (related to Proposition: 1.05: Isosceles Triangles I)
        
        Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I) (1)
        
        Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
        
        Proposition: 1.06: Isosceles Triagles II (2)
        
        Proof: By Euclid (related to Proposition: 1.06: Isosceles Triagles II)
        
        Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II) (1)
        
        Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
        
        Proposition: 1.07: Uniqueness of Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.07: Uniqueness of Triangles)
        
        Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles)
        
        Proposition: 1.09: Bisecting an Angle (1)
        
        Proof: By Euclid (related to Proposition: 1.09: Bisecting an Angle)
        
        Proposition: 1.10: Bisecting a Segment (1)
        
        Proof: By Euclid (related to Proposition: 1.10: Bisecting a Segment)
        
        Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line (1)
        
        Proof: (Euclid) (related to Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line)
        
        Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line (1)
        
        Proof: (Euclid) (related to Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line)
        
        Proposition: 1.13: Angles at Intersections of Straight Lines (3)
        
        Proof: By Euclid (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
        
        Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines) (1)
        
        Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
        
        Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines) (1)
        
        Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
        
        Proposition: 1.14: Combining Rays to Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 1.14: Combining Rays to Straight Lines)
        
        Proposition: 1.15: Opposite Angles on Intersecting Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 1.15: Opposite Angles on Intersecting Straight Lines)
        
        Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles (2)
        
        Proof: By Euclid (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
        
        Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles) (1)
        
        Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
        
        Proposition: 1.17: The Sum of Two Angles of a Triangle (1)
        
        Proof: By Euclid (related to Proposition: 1.17: The Sum of Two Angles of a Triangle)
        
        Proposition: 1.18: Angles and Sides in a Triangle I (1)
        
        Proof: (Euclid) (related to Proposition: 1.18: Angles and Sides in a Triangle I)
        
        Proposition: 1.19: Angles and Sides in a Triangle II (1)
        
        Proof: (Euclid) (related to Proposition: 1.19: Angles and Sides in a Triangle II)
        
        Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality) (1)
        
        Proof: By Euclid (related to Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality))
        
        Proposition: 1.21: Triangles within Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.21: Triangles within Triangles)
        
        Proposition: 1.22: Construction of Triangles From Arbitrary Segments (1)
        
        Proof: By Euclid (related to Proposition: 1.22: Construction of Triangles From Arbitrary Segments)
        
        Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle (1)
        
        Proof: By Euclid (related to Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle)
        
        Proposition: 1.24: Angles and Sides in a Triangle III (1)
        
        Proof: (Euclid) (related to Proposition: 1.24: Angles and Sides in a Triangle III)
        
        Proposition: 1.25: Angles and Sides in a Triangle IV (2)
        
        Proof: (Euclid) (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
        
        Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV) (1)
        
        Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
        
        Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles (1)
        
        Proof: By Euclid (related to Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles)
        
        Proposition: 1.27: Parallel Lines I (1)
        
        Proof: By Euclid (related to Proposition: 1.27: Parallel Lines I)
        
        Proposition: 1.28: Parallel Lines II (1)
        
        Proof: By Euclid (related to Proposition: 1.28: Parallel Lines II)
        
        Proposition: 1.29: Parallel Lines III (2)
        
        Proof: By Euclid (related to Proposition: 1.29: Parallel Lines III)
        
        Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III) (1)
        
        Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
        
        Proposition: 1.30: Transitivity of Parallel Lines (1)
        
        Proof: By Euclid (related to Proposition: 1.30: Transitivity of Parallel Lines)
        
        Proposition: 1.31: Constructing a Parallel Line from a Line and a Point (1)
        
        Proof: (Euclid) (related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)
        
        Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle (6)
        
        Proof: By Euclid (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
        
        Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
        
        Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Angles of Right Triangle)
        
        Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
        
        Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
        
        Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
        
        Proof: By Euclid (related to Corollary: Similar Triangles)
        
        Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram (1)
        
        Proof: By Euclid (related to Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram)
        
        Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms (8)
        
        Proof: By Euclid (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
        
        Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
        
        Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
        
        Definition: Parallelogram - Defining Property III
        
        Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Rhombus as a Special Case of a Parallelogram)
        
        Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
        
        Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
        
        Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
        
        Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
        
        Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels (1)
        
        Proof: By Euclid (related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
        
        Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels (1)
        
        Proof: By Euclid (related to Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels)
        
        Proposition: 1.37: Triangles of Equal Area I (1)
        
        Proof: By Euclid (related to Proposition: 1.37: Triangles of Equal Area I)
        
        Proposition: 1.38: Triangles of Equal Area II (1)
        
        Proof: By Euclid (related to Proposition: 1.38: Triangles of Equal Area II)
        
        Proposition: 1.39: Triangles of Equal Area III (1)
        
        Proof: By Euclid (related to Proposition: 1.39: Triangles of Equal Area III)
        
        Proposition: 1.40: Triangles of Equal Area IV (1)
        
        Proof: By Euclid (related to Proposition: 1.40: Triangles of Equal Area IV)
        
        Proposition: 1.41: Parallelograms and Triagles (1)
        
        Proof: By Euclid (related to Proposition: 1.41: Parallelograms and Triagles)
        
        Proposition: 1.42: Construction of Parallelograms I (1)
        
        Proof: By Euclid (related to Proposition: 1.42: Construction of Parallelograms I)
        
        Proposition: 1.43: Complementary Segments of Parallelograms (1)
        
        Proof: (Euclid) (related to Proposition: 1.43: Complementary Segments of Parallelograms)
        
        Proposition: 1.44: Construction of Parallelograms II (1)
        
        Proof: By Euclid (related to Proposition: 1.44: Construction of Parallelograms II)
        
        Proposition: 1.45: Construction of Parallelograms III (1)
        
        Proof: By Euclid (related to Proposition: 1.45: Construction of Parallelograms III)
        
        Proposition: 1.46: Construction of a Square on a Given Segment (2)
        
        Proof: By Euclid (related to Proposition: 1.46: Construction of a Square on a Given Segment)
        
        Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment) (1)
        
        Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
        
        Proposition: 1.47: Pythagorean Theorem (1)
        
        Proof: By Euclid (related to Proposition: 1.47: Pythagorean Theorem)
        
        Proposition: 1.48: The Converse of the Pythagorean Theorem (1)
        
        Proof: By Euclid (related to Proposition: 1.48: The Converse of the Pythagorean Theorem)
      - Section: Book 02: Fundamentals of Geometric Algebra (16)
        
        Subsection: Definitions from Book 2 (2)
        
        Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides (1)
        
        Definition: Point of Division, Point of External Division
        
        Definition: 2.2: Gnomon
        
        Subsection: Propositions from Book 2 (14)
        
        Proposition: 2.01: Summing Areas or Rectangles (1)
        
        Proof: By Euclid (related to Proposition: 2.01: Summing Areas or Rectangles)
        
        Proposition: 2.02: Square is Sum of Two Rectangles (1)
        
        Proof: By Euclid (related to Proposition: 2.02: Square is Sum of Two Rectangles)
        
        Proposition: 2.03: Rectangle is Sum of Square and Rectangle (1)
        
        Proof: By Euclid (related to Proposition: 2.03: Rectangle is Sum of Square and Rectangle)
        
        Proposition: 2.04: Square of Sum (1)
        
        Proof: By Euclid (related to Proposition: 2.04: Square of Sum)
        
        Proposition: 2.05: Rectangle is Difference of Two Squares (1)
        
        Proof: By Euclid (related to Proposition: 2.05: Rectangle is Difference of Two Squares)
        
        Proposition: 2.06: Square of Sum with One Halved Summand (1)
        
        Proof: By Euclid (related to Proposition: 2.06: Square of Sum with One Halved Summand)
        
        Proposition: 2.07: Sum of Squares (1)
        
        Proof: By Euclid (related to Proposition: 2.07: Sum of Squares)
        
        Proposition: 2.08: Square of Sum with One Doubled Summand (1)
        
        Proof: By Euclid (related to Proposition: 2.08: Square of Sum with One Doubled Summand)
        
        Proposition: 2.09: Sum of Squares of Sum and Difference (1)
        
        Proof: By Euclid (related to Proposition: 2.09: Sum of Squares of Sum and Difference)
        
        Proposition: 2.10: Sum of Squares (Half) (1)
        
        Proof: By Euclid (related to Proposition: 2.10: Sum of Squares (Half))
        
        Proposition: 2.11: Constructing the Golden Ratio of a Segment (1)
        
        Proof: By Euclid (related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)
        
        Proposition: 2.12: Law of Cosines (for Obtuse Angles) (1)
        
        Proof: By Euclid (related to Proposition: 2.12: Law of Cosines (for Obtuse Angles))
        
        Proposition: 2.13: Law of Cosines (for Acute Angles) (1)
        
        Proof: By Euclid (related to Proposition: 2.13: Law of Cosines (for Acute Angles))
        
        Proposition: 2.14: Constructing a Square from a Rectilinear Figure (1)
        
        Proof: By Euclid (related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)
      - Section: Book 03: Fundamentals of Plane Geometry Involving Circles (50)
        
        Subsection: Definitions from Book 3 (11)
        
        Definition: 3.01: Congruent Circles
        
        Definition: 3.02: Tangent to the Circle, Straight Line Touching The Circle
        
        Definition: 3.03: Circles Touching One Another
        
        Definition: 3.04: Chords Equally Far From the Center of a Circle
        
        Definition: 3.05: Chords Being Further from the Center of a Circle
        
        Definition: 3.06: Segment of a Circle, Arc
        
        Definition: 3.07: Angle of a Segment
        
        Definition: 3.08: Angle in the Segment (Inscribed Angle)
        
        Definition: 3.09: Angle Standing Upon An Arc
        
        Definition: 3.10: Circular Sector, Central Angle
        
        Definition: 3.11: Similar Circular Segments
        
        Subsection: Propositions from Book 3 (39)
        
        Proposition: 3.01: Finding the Center of a given Circle (2)
        
        Proof: By Euclid (related to Proposition: 3.01: Finding the Center of a given Circle)
        
        Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle) (1)
        
        Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
        
        Proposition: 3.02: Chord Lies Inside its Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.02: Chord Lies Inside its Circle)
        
        Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector (1)
        
        Proof: By Euclid (related to Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector)
        
        Proposition: 3.04: Chords do not Bisect Each Other (1)
        
        Proof: By Euclid (related to Proposition: 3.04: Chords do not Bisect Each Other)
        
        Proposition: 3.05: Intersecting Circles have Different Centers (1)
        
        Proof: By Euclid (related to Proposition: 3.05: Intersecting Circles have Different Centers)
        
        Proposition: 3.06: Touching Circles have Different Centers (1)
        
        Proof: By Euclid (related to Proposition: 3.06: Touching Circles have Different Centers)
        
        Proposition: 3.07: Relative Lengths of Lines Inside Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.07: Relative Lengths of Lines Inside Circle)
        
        Proposition: 3.08: Relative Lengths of Lines Outside Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.08: Relative Lengths of Lines Outside Circle)
        
        Proposition: 3.09: Condition for Point to be Center of Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.09: Condition for Point to be Center of Circle)
        
        Proposition: 3.10: Two Circles have at most Two Points of Intersection (1)
        
        Proof: By Euclid (related to Proposition: 3.10: Two Circles have at most Two Points of Intersection)
        
        Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally (1)
        
        Proof: By Euclid (related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
        
        Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally (1)
        
        Proof: By Euclid (related to Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally)
        
        Proposition: 3.13: Circles Touch at One Point at Most (1)
        
        Proof: By Euclid (related to Proposition: 3.13: Circles Touch at One Point at Most)
        
        Proposition: 3.14: Equal Chords in Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.14: Equal Chords in Circle)
        
        Proposition: 3.15: Relative Lengths of Chords of Circles (1)
        
        Proof: By Euclid (related to Proposition: 3.15: Relative Lengths of Chords of Circles)
        
        Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (2)
        
        Proof: By Euclid (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
        
        Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle) (1)
        
        Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
        
        Proposition: 3.17: Construction of Tangent from Point to Circle (1)
        
        Proof: By Euclid (related to Proposition: 3.17: Construction of Tangent from Point to Circle)
        
        Proposition: 3.18: Radius at Right Angle to Tangent (1)
        
        Proof: By Euclid (related to Proposition: 3.18: Radius at Right Angle to Tangent)
        
        Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center (1)
        
        Proof: By Euclid (related to Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center)
        
        Proposition: 3.20: Inscribed Angle Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.20: Inscribed Angle Theorem)
        
        Proposition: 3.21: Angles in Same Segment of Circle are Equal (1)
        
        Proof: By Euclid (related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
        
        Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral (1)
        
        Proof: By Euclid (related to Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral)
        
        Proposition: 3.23: Segment on Given Base Unique (1)
        
        Proof: By Euclid (related to Proposition: 3.23: Segment on Given Base Unique)
        
        Proposition: 3.24: Similar Segments on Equal Bases are Equal (1)
        
        Proof: By Euclid (related to Proposition: 3.24: Similar Segments on Equal Bases are Equal)
        
        Proposition: 3.25: Construction of Circle from Segment (1)
        
        Proof: By Euclid (related to Proposition: 3.25: Construction of Circle from Segment)
        
        Proposition: 3.26: Equal Angles and Arcs in Equal Circles (1)
        
        Proof: By Euclid (related to Proposition: 3.26: Equal Angles and Arcs in Equal Circles)
        
        Proposition: 3.27: Angles on Equal Arcs are Equal (1)
        
        Proof: By Euclid (related to Proposition: 3.27: Angles on Equal Arcs are Equal)
        
        Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles (1)
        
        Proof: By Euclid (related to Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles)
        
        Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
        
        Proposition: 3.30: Bisection of Arc (1)
        
        Proof: By Euclid (related to Proposition: 3.30: Bisection of Arc)
        
        Proposition: 3.31: Relative Sizes of Angles in Segments (1)
        
        Proof: By Euclid (related to Proposition: 3.31: Relative Sizes of Angles in Segments)
        
        Proposition: 3.32: Angles made by Chord with Tangent (1)
        
        Proof: By Euclid (related to Proposition: 3.32: Angles made by Chord with Tangent)
        
        Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle (1)
        
        Proof: By Euclid (related to Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle)
        
        Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle (1)
        
        Proof: By Euclid (related to Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle)
        
        Proposition: 3.35: Intersecting Chord Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.35: Intersecting Chord Theorem)
        
        Proposition: 3.36: Tangent Secant Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.36: Tangent Secant Theorem)
        
        Proposition: 3.37: Converse of Tangent Secant Theorem (1)
        
        Proof: By Euclid (related to Proposition: 3.37: Converse of Tangent Secant Theorem)
      - Section: Book 04: Circles: Inscription and Circumscription (24)
        
        Subsection: Definitions from Book 4 (7)
        
        Definition: 4.1: Rectilinear Figure Inscribed in Another Rectilinear Figure
        
        Definition: 4.2: Rectilinear Figure Circumscribed about Another Rectilinear Figure
        
        Definition: 4.3: Inscribing Rectilinear Figures in Circles
        
        Definition: 4.4: Circumscribing Rectilinear Figures about Circles
        
        Definition: 4.7: Chord and Secant
        
        Definition: 4.6: Circumscribing Circles about Rectilinear Figures
        
        Definition: 4.5: Inscribing Circles in Rectilinear Figures
        
        Subsection: Propositions from Book 4 (17)
        
        Proposition: 4.01: Fitting Chord Into Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.01: Fitting Chord Into Circle)
        
        Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles (1)
        
        Proof: By Euclid (related to Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles)
        
        Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles (1)
        
        Proof: By Euclid (related to Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles)
        
        Proposition: 4.04: Inscribing Circle in Triangle (1)
        
        Proof: By Euclid (related to Proposition: 4.04: Inscribing Circle in Triangle)
        
        Proposition: 4.05: Circumscribing Circle about Triangle (1)
        
        Proof: By Euclid (related to Proposition: 4.05: Circumscribing Circle about Triangle)
        
        Proposition: 4.06: Inscribing Square in Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.06: Inscribing Square in Circle)
        
        Proposition: 4.07: Circumscribing Square about Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.07: Circumscribing Square about Circle)
        
        Proposition: 4.08: Inscribing Circle in Square (1)
        
        Proof: By Euclid (related to Proposition: 4.08: Inscribing Circle in Square)
        
        Proposition: 4.09: Circumscribing Circle about Square (1)
        
        Proof: By Euclid (related to Proposition: 4.09: Circumscribing Circle about Square)
        
        Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex (1)
        
        Proof: By Euclid (related to Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex)
        
        Proposition: 4.11: Inscribing Regular Pentagon in Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.11: Inscribing Regular Pentagon in Circle)
        
        Proposition: 4.12: Circumscribing Regular Pentagon about Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.12: Circumscribing Regular Pentagon about Circle)
        
        Proposition: 4.13: Inscribing Circle in Regular Pentagon (1)
        
        Proof: By Euclid (related to Proposition: 4.13: Inscribing Circle in Regular Pentagon)
        
        Proposition: 4.14: Circumscribing Circle about Regular Pentagon (1)
        
        Proof: By Euclid (related to Proposition: 4.14: Circumscribing Circle about Regular Pentagon)
        
        Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (2)
        
        Proof: By Euclid (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
        
        Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle) (1)
        
        Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
        
        Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle (1)
        
        Proof: By Euclid (related to Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle)
      - Section: Book 05: Proportion (45)
        
        Subsection: Definitions from Book 5 (18)
        
        Definition: 5.01: Magnitude is Aliquot Part
        
        Definition: 5.02: Multiple of a Real Number
        
        Definition: 5.03: Ratio
        
        Definition: 5.04: Having a Ratio
        
        Definition: 5.05: Having the Same Ratio
        
        Definition: 5.06: Proportional Magnitudes (1)
        
        Definition: Geometric Progression, Continued Proportion
        
        Definition: 5.07: Having a Greater Ratio
        
        Definition: 5.08: Proportion in Three Terms
        
        Definition: 5.09: Squared Ratio
        
        Definition: 5.10: Cubed Ratio
        
        Definition: 5.11: Corresponding Magnitudes
        
        Definition: 5.12: Alternate Ratio
        
        Definition: 5.13: Inverse Ratio
        
        Definition: 5.14: Composition of a Ratio
        
        Definition: 5.15: Separation of a Ratio
        
        Definition: 5.16: Conversion of a Ratio
        
        Definition: 5.17: Ratio ex Aequali
        
        Definition: 5.18: Perturbed Proportion
        
        Subsection: Propositions from Book 5 (27)
        
        Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition (1)
        
        Proof: By Euclid (related to Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition)
        
        Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition (1)
        
        Proof: By Euclid (related to Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition)
        
        Proposition: 5.03: Multiplication of Numbers is Associative (1)
        
        Proof: By Euclid (related to Proposition: 5.03: Multiplication of Numbers is Associative)
        
        Proposition: 5.04: Multiples of Terms in Equal Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.04: Multiples of Terms in Equal Ratios)
        
        Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction (1)
        
        Proof: By Euclid (related to Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction)
        
        Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction (1)
        
        Proof: By Euclid (related to Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction)
        
        Proposition: 5.07: Ratios of Equal Magnitudes (2)
        
        Proof: By Euclid (related to Proposition: 5.07: Ratios of Equal Magnitudes)
        
        Corollary: 5.07: Ratios of Equal Magnitudes (related to Proposition: 5.07: Ratios of Equal Magnitudes) (1)
        
        Proof: By Euclid (related to Corollary: 5.07: Ratios of Equal Magnitudes)
        
        Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes (1)
        
        Proof: By Euclid (related to Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes)
        
        Proposition: 5.09: Magnitudes with Same Ratios are Equal (1)
        
        Proof: By Euclid (related to Proposition: 5.09: Magnitudes with Same Ratios are Equal)
        
        Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios)
        
        Proposition: 5.11: Equality of Ratios is Transitive (1)
        
        Proof: By Euclid (related to Proposition: 5.11: Equality of Ratios is Transitive)
        
        Proposition: 5.12: Sum of Components of Equal Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.12: Sum of Components of Equal Ratios)
        
        Proposition: 5.13: Relative Sizes of Proportional Magnitudes (1)
        
        Proof: By Euclid (related to Proposition: 5.13: Relative Sizes of Proportional Magnitudes)
        
        Proposition: 5.14: Relative Sizes of Components of Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.14: Relative Sizes of Components of Ratios)
        
        Proposition: 5.15: Ratio Equals its Multiples (1)
        
        Proof: By Euclid (related to Proposition: 5.15: Ratio Equals its Multiples)
        
        Proposition: 5.16: Proportional Magnitudes are Proportional Alternately (1)
        
        Proof: By Euclid (related to Proposition: 5.16: Proportional Magnitudes are Proportional Alternately)
        
        Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated (1)
        
        Proof: By Euclid (related to Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated)
        
        Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded (1)
        
        Proof: By Euclid (related to Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded)
        
        Proposition: 5.19: Proportional Magnitudes have Proportional Remainders (2)
        
        Proof: By Euclid (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
        
        Corollary: 5.19: Proportional Magnitudes have Proportional Remainders (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders) (1)
        
        Proof: By Euclid (related to Corollary: 5.19: Proportional Magnitudes have Proportional Remainders)
        
        Proposition: 5.20: Relative Sizes of Successive Ratios (1)
        
        Proof: By Euclid (related to Proposition: 5.20: Relative Sizes of Successive Ratios)
        
        Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion)
        
        Proposition: 5.22: Equality of Ratios Ex Aequali (1)
        
        Proof: By Euclid (related to Proposition: 5.22: Equality of Ratios Ex Aequali)
        
        Proposition: 5.23: Equality of Ratios in Perturbed Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.23: Equality of Ratios in Perturbed Proportion)
        
        Proposition: 5.24: Sum of Antecedents of Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.24: Sum of Antecedents of Proportion)
        
        Proposition: 5.25: Sum of Antecedent and Consequent of Proportion (1)
        
        Proof: By Euclid (related to Proposition: 5.25: Sum of Antecedent and Consequent of Proportion)
      - Section: Book 06: Similar Figures (39)
        
        Subsection: Definitions from Book 6 (3)
        
        Definition: 6.01: Similar Rectilinear Figures
        
        Definition: 6.02: Cut in Extreme and Mean Ratio
        
        Definition: 6.03: Height of a Figure
        
        Subsection: Propositions from Book 6 (36)
        
        Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base (1)
        
        Proof: By Euclid (related to Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base)
        
        Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally (1)
        
        Proof: By Euclid (related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
        
        Proposition: 6.03: Angle Bisector Theorem (1)
        
        Proof: By Euclid (related to Proposition: 6.03: Angle Bisector Theorem)
        
        Proposition: 6.04: Equiangular Triangles are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.04: Equiangular Triangles are Similar)
        
        Proposition: 6.05: Triangles with Proportional Sides are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.05: Triangles with Proportional Sides are Similar)
        
        Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar)
        
        Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar)
        
        Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles (2)
        
        Proof: By Euclid (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
        
        Corollary: 6.08: Geometric Mean Theorem (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles) (1)
        
        Proof: By Euclid (related to Corollary: 6.08: Geometric Mean Theorem)
        
        Proposition: 6.09: Construction of Part of Line (1)
        
        Proof: By Euclid (related to Proposition: 6.09: Construction of Part of Line)
        
        Proposition: 6.10: Construction of Similarly Cut Straight Line (1)
        
        Proof: By Euclid (related to Proposition: 6.10: Construction of Similarly Cut Straight Line)
        
        Proposition: 6.11: Construction of Segment in Squared Ratio (1)
        
        Proof: By Euclid (related to Proposition: 6.11: Construction of Segment in Squared Ratio)
        
        Proposition: 6.12: Construction of Fourth Proportional Straight Line (1)
        
        Proof: By Euclid (related to Proposition: 6.12: Construction of Fourth Proportional Straight Line)
        
        Proposition: 6.13: Construction of Mean Proportional (1)
        
        Proof: By Euclid (related to Proposition: 6.13: Construction of Mean Proportional)
        
        Proposition: 6.14: Characterization of Congruent Parallelograms (1)
        
        Proof: By Euclid (related to Proposition: 6.14: Characterization of Congruent Parallelograms)
        
        Proposition: 6.15: Characterization of Congruent Triangles (1)
        
        Proof: By Euclid (related to Proposition: 6.15: Characterization of Congruent Triangles)
        
        Proposition: 6.16: Rectangles Contained by Proportional Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 6.16: Rectangles Contained by Proportional Straight Lines)
        
        Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)
        
        Proposition: 6.18: Construction of Similar Polygon (1)
        
        Proof: By Euclid (related to Proposition: 6.18: Construction of Similar Polygon)
        
        Proposition: 6.19: Ratio of Areas of Similar Triangles (2)
        
        Proof: By Euclid (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
        
        Corollary: 6.19: Ratio of Areas of Similar Triangles (related to Proposition: 6.19: Ratio of Areas of Similar Triangles) (1)
        
        Proof: By Euclid (related to Corollary: 6.19: Ratio of Areas of Similar Triangles)
        
        Proposition: 6.20: Similar Polygons are Composed of Similar Triangles (2)
        
        Proof: By Euclid (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
        
        Corollary: 6.20: Similar Polygons are Composed of Similar Triangles (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles) (1)
        
        Proof: By Euclid (related to Corollary: 6.20: Similar Polygons are Composed of Similar Triangles)
        
        Proposition: 6.21: Similarity of Polygons is Transitive (1)
        
        Proof: By Euclid (related to Proposition: 6.21: Similarity of Polygons is Transitive)
        
        Proposition: 6.22: Similar Figures on Proportional Straight Lines (1)
        
        Proof: By Euclid (related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)
        
        Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms (1)
        
        Proof: By Euclid (related to Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms)
        
        Proposition: 6.24: Parallelograms About Diameter are Similar (1)
        
        Proof: By Euclid (related to Proposition: 6.24: Parallelograms About Diameter are Similar)
        
        Proposition: 6.25: Construction of Figure Similar to One and Equal to Another (1)
        
        Proof: By Euclid (related to Proposition: 6.25: Construction of Figure Similar to One and Equal to Another)
        
        Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter (1)
        
        Proof: By Euclid (related to Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter)
        
        Proposition: 6.27: Similar Parallelogram on Half a Straight Line (1)
        
        Proof: By Euclid (related to Proposition: 6.27: Similar Parallelogram on Half a Straight Line)
        
        Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram (1)
        
        Proof: By Euclid (related to Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram)
        
        Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram (1)
        
        Proof: By Euclid (related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)
        
        Proposition: 6.30: Construction of the Inverse Golden Section (1)
        
        Proof: By Euclid (related to Proposition: 6.30: Construction of the Inverse Golden Section)
        
        Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle (1)
        
        Proof: By Euclid (related to Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle)
        
        Proposition: 6.32: Triangles with Two Sides Parallel and Equal (1)
        
        Proof: By Euclid (related to Proposition: 6.32: Triangles with Two Sides Parallel and Equal)
        
        Proposition: 6.33: Angles in Circles have Same Ratio as Arcs (1)
        
        Proof: By Euclid (related to Proposition: 6.33: Angles in Circles have Same Ratio as Arcs)
      - Section: Book 07: Elementary Number Theory (62)
        
        Subsection: Definitions from Book 7 (22)
        
        Definition: 7.01: Unit
        
        Definition: 7.02: Number
        
        Definition: 7.03: Proper Divisor
        
        Definition: 7.04: Aliquant Part, a Number Being Not a Divisor of Another Number
        
        Definition: 7.05: Multiple, Number Multiplying another Number
        
        Definition: 7.06: Even Number
        
        Definition: 7.07: Odd Number
        
        Definition: 7.08: Even-Times-Even Number
        
        Definition: 7.09: Even-Times-Odd Number
        
        Definition: 7.10: Odd-Times-Odd Number
        
        Definition: 7.11: Prime Number
        
        Definition: 7.12: Co-prime (Relatively Prime) Numbers
        
        Definition: 7.13: Composite Number
        
        Definition: 7.14: Not Co-prime Numbers
        
        Definition: 7.15: Multiplication of Numbers
        
        Definition: 7.16: Rectangular Number, Plane Number
        
        Definition: 7.17: Cuboidal Number, Solid Number
        
        Definition: 7.18: Square Number
        
        Definition: 7.19: Cubic Number, Cube Number
        
        Definition: 7.20: Proportional Numbers
        
        Definition: 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers
        
        Definition: 7.22: Perfect Number
        
        Subsection: Propositions from Book 7 (40)
        
        Proposition: 7.01: Sufficient Condition for Coprimality (1)
        
        Proof: By Euclid (related to Proposition: 7.01: Sufficient Condition for Coprimality)
        
        Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm (2)
        
        Proof: By Euclid (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
        
        Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm) (1)
        
        Proof: By Euclid (related to Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor)
        
        Proposition: 7.03: Greatest Common Divisor of Three Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)
        
        Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers)
        
        Proposition: 7.05: Divisors Obey Distributive Law (Sum) (1)
        
        Proof: By Euclid (related to Proposition: 7.05: Divisors Obey Distributive Law (Sum))
        
        Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum) (1)
        
        Proof: By Euclid (related to Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum))
        
        Proposition: 7.07: Divisors Obey Distributive Law (Difference) (1)
        
        Proof: By Euclid (related to Proposition: 7.07: Divisors Obey Distributive Law (Difference))
        
        Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference) (1)
        
        Proof: By Euclid (related to Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference))
        
        Proposition: 7.09: Alternate Ratios of Equal Fractions (1)
        
        Proof: By Euclid (related to Proposition: 7.09: Alternate Ratios of Equal Fractions)
        
        Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions (1)
        
        Proof: By Euclid (related to Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions)
        
        Proposition: 7.11: Proportional Numbers have Proportional Differences (1)
        
        Proof: By Euclid (related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
        
        Proposition: 7.12: Ratios of Numbers is Distributive over Addition (1)
        
        Proof: By Euclid (related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
        
        Proposition: 7.13: Proportional Numbers are Proportional Alternately (1)
        
        Proof: By Euclid (related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
        
        Proposition: 7.14: Proportion of Numbers is Transitive (1)
        
        Proof: By Euclid (related to Proposition: 7.14: Proportion of Numbers is Transitive)
        
        Proposition: 7.15: Alternate Ratios of Multiples (1)
        
        Proof: By Euclid (related to Proposition: 7.15: Alternate Ratios of Multiples)
        
        Proposition: 7.16: Natural Number Multiplication is Commutative (1)
        
        Proof: By Euclid (related to Proposition: 7.16: Natural Number Multiplication is Commutative)
        
        Proposition: 7.17: Multiples of Ratios of Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.17: Multiples of Ratios of Numbers)
        
        Proposition: 7.18: Ratios of Multiples of Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.18: Ratios of Multiples of Numbers)
        
        Proposition: 7.19: Relation of Ratios to Products (1)
        
        Proof: By Euclid (related to Proposition: 7.19: Relation of Ratios to Products)
        
        Proposition: 7.20: Ratios of Fractions in Lowest Terms (1)
        
        Proof: By Euclid (related to Proposition: 7.20: Ratios of Fractions in Lowest Terms)
        
        Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms (1)
        
        Proof: By Euclid (related to Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms)
        
        Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime)
        
        Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other (1)
        
        Proof: By Euclid (related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
        
        Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole (1)
        
        Proof: By Euclid (related to Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole)
        
        Proposition: 7.25: Square of Co-prime Number is Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.25: Square of Co-prime Number is Co-prime)
        
        Proposition: 7.26: Product of Co-prime Pairs is Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.26: Product of Co-prime Pairs is Co-prime)
        
        Proposition: 7.27: Powers of Co-prime Numbers are Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.27: Powers of Co-prime Numbers are Co-prime)
        
        Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both (1)
        
        Proof: By Euclid (related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)
        
        Proposition: 7.29: Prime not Divisor implies Co-prime (1)
        
        Proof: By Euclid (related to Proposition: 7.29: Prime not Divisor implies Co-prime)
        
        Proposition: 7.30: Euclidean Lemma (1)
        
        Proof: By Euclid (related to Proposition: 7.30: Euclidean Lemma)
        
        Proposition: 7.31: Existence of Prime Divisors (1)
        
        Proof: By Euclid (related to Proposition: 7.31: Existence of Prime Divisors)
        
        Proposition: 7.32: Natural Number is Prime or has Prime Factor (1)
        
        Proof: By Euclid (related to Proposition: 7.32: Natural Number is Prime or has Prime Factor)
        
        Proposition: 7.33: Least Ratio of Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.33: Least Ratio of Numbers)
        
        Proposition: 7.34: Existence of Least Common Multiple (1)
        
        Proof: By Euclid (related to Proposition: 7.34: Existence of Least Common Multiple)
        
        Proposition: 7.35: Least Common Multiple Divides Common Multiple (1)
        
        Proof: By Euclid (related to Proposition: 7.35: Least Common Multiple Divides Common Multiple)
        
        Proposition: 7.36: Least Common Multiple of Three Numbers (1)
        
        Proof: By Euclid (related to Proposition: 7.36: Least Common Multiple of Three Numbers)
        
        Proposition: 7.37: Integer Divided by Divisor is Integer (1)
        
        Proof: By Euclid (related to Proposition: 7.37: Integer Divided by Divisor is Integer)
        
        Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer (1)
        
        Proof: By Euclid (related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
        
        Proposition: 7.39: Least Number with Three Given Fractions (1)
        
        Proof: By Euclid (related to Proposition: 7.39: Least Number with Three Given Fractions)
      - Section: Book 08: Continued Proportion (28)
        
        Subsection: Propositions from Book 8 (28)
        
        Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms (1)
        
        Proof: By Euclid (related to Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms)
        
        Proposition: 8.02: Construction of Geometric Progression in Lowest Terms (2)
        
        Proof: By Euclid (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
        
        Corollary: 8.02: Construction of Geometric Progression in Lowest Terms (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms) (1)
        
        Proof: By Euclid (related to Corollary: 8.02: Construction of Geometric Progression in Lowest Terms)
        
        Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes (1)
        
        Proof: By Euclid (related to Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes)
        
        Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios (1)
        
        Proof: By Euclid (related to Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios)
        
        Proposition: 8.05: Ratio of Products of Sides of Plane Numbers (1)
        
        Proof: By Euclid (related to Proposition: 8.05: Ratio of Products of Sides of Plane Numbers)
        
        Proposition: 8.06: First Element of Geometric Progression not dividing Second (1)
        
        Proof: By Euclid (related to Proposition: 8.06: First Element of Geometric Progression not dividing Second)
        
        Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second (1)
        
        Proof: By Euclid (related to Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second)
        
        Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements (1)
        
        Proof: By Euclid (related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
        
        Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers)
        
        Proposition: Prop. 8.10: Product of Geometric Progressions from One (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.10: Product of Geometric Progressions from One)
        
        Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional)
        
        Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals)
        
        Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression)
        
        Proposition: Prop. 8.14: Number divides Number iff Square divides Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.14: Number divides Number iff Square divides Square)
        
        Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube)
        
        Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
        
        Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
        
        Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional)
        
        Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals)
        
        Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane)
        
        Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid)
        
        Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)
        
        Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube)
        
        Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)
        
        Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube)
        
        Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares (1)
        
        Proof: By Euclid (related to Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares)
        
        Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes (1)
        
        Proof:

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Index: Content Tree