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)
- 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)
- 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)
- Definition: Quantifier, Bound Variables, Free Variables (1)
- Definition: Function, Arity and Constant (1)
- Definition: Predicate of a Logical Calculus (1)
- Definition: Signature

- Chapter: Putting it All Together - Syntax and Semantics of a Logical Calculus (3)
- 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)
- Definition: Semantics of PL0 (15)
- Definition: `$k$`-nary Connectives, Prime and Compound Propositions
- Lemma: Boolean Function (2)
- Definition: Truth Table
- Definition: Negation
- Definition: Conjunction (2)
- Definition: Disjunction (3)
- Corollary: Commutativity of Disjunction (related to Definition: Disjunction) (1)
- Proposition: Associativity of Disjunction (1)
- Definition: Exclusive Disjunction

- Definition: Implication (4)
- Lemma: Implication as a Disjunction (1)
- Lemma: Negation of an Implication (2)
- Definition: Contrapositive

- Definition: Equivalence (1)

- Chapter: Equivalent Transformations in Propositional Logic (5)
- Lemma: Every Proposition Implies Itself (1)
- Lemma: It is true that something can be (either) true or false (1)
- Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition (1)
- Lemma: De Morgan's Laws (Logic) (1)
- Lemma: Distributivity of Conjunction and Disjunction (1)

- Chapter: Contradictory Propositions in Propositional Logic (2)
- Chapter: Normal Forms in `$PL0$` (8)
- Section: Examples of Propositions With Different Syntactic Forms but the Same Boolean Function
- Definition: Canonical Normal Form
- Definition: Literals, Minterms, and Maxterms
- Lemma: Unique Valuation of Minterms and Maxterms (1)
- Definition: Conjunctive and Disjunctive Canonical Normal Forms
- Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms (2)
- Example: Examples of Canonical Normal Forms (related to Chapter: Normal Forms in `$PL0$`)

- Definition: Boolean Algebra (1)

- 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)
- Lemma: Mixing-up the Sufficient and Necessary Conditions (1)
- Lemma: Affirming the Consequent of an Implication (1)
- Lemma: Denying the Antecedent of an Implication (1)

- 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)
- Lemma: Modus Ponens (1)
- Lemma: Modus Tollens (1)
- Lemma: Hypothetical Syllogism (1)
- Lemma: Disjunctive Syllogism (1)
- Lemma: The Proving Principle by Contradiction (1)
- Lemma: The Proving Principle By Contraposition, Contrapositive (1)
- Lemma: The Proving Principle by Complete Induction (2)
- Lemma: The Proving Principle by Transfinite Induction (1)

- 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)
- 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)
- Proposition: Set Union is Commutative (1)
- Proposition: Set Union is Associative (1)

- Definition: Set Intersection (4)
- Definition: Disjoint Sets
- Proposition: Intersection of a Set With Another Set is Subset of This Set (1)
- Proposition: Set Intersection is Commutative (1)
- Proposition: Set Intersection is Associative (1)

- Definition: Set Complement
- Definition: Set Difference
- Proposition: De Morgan's Laws (Sets) (1)
- Proposition: Distributivity Laws For Sets (1)
- Proposition: Sets and Their Complements (1)

- Definition: Set Union (3)
- 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)
- Axiom: Axiom of Extensionality (1)
- 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)
- Corollary: Justification of Subsets and Supersets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
- Corollary: Equality of Sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
- 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)
- Corollary: Justification of Set Intersection (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
- Corollary: Justification of the Difference (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)
- Corollary: Set Difference and Set Complement are the Same Concepts (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)) (1)

- Axiom: Axiom of Pairing
- Axiom: Axiom of Union (1)
- Axiom: Axiom of Power Set (2)
- Corollary: Justification of Power Set (related to Axiom: Axiom of Power Set) (1)
- Definition: Singleton

- Axiom: Axiom of Foundation (1)
- Axiom: Axiom of Infinity (3)
- Axiom: Axiom of Replacement (Schema)
- Axiom: Axiom of Choice

- Axiom: Zermelo-Fraenkel Axioms (21)
- Part: Relations (60)
- Definition: Ordered Pair, n-Tuple (1)
- Motivation: Usage of Ordered Tuples In Other Mathematical Disciplines (related to Part: Relations)
- Definition: Cartesian Product
- Definition: Relation (1)
- 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)
- 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)
- 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)
- Section: Important Properties of Functions (6)
- Definition: Injective Function
- Definition: Surjective Function
- Definition: Bijective Function
- Proposition: Characterization of Bijective Functions (1)
- Definition: Invertible Functions, Inverse Functions
- Proposition: Functions Constitute Equivalence Relations (1)

- 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)
- Lemma: Composition of Functions (7)
- Proof: (related to Lemma: Composition of Functions)
- Proposition: Composition of Functions is Associative (1)
- Proposition: Composition of Surjective Functions is Surjective (1)
- Proposition: Composition of Injective Functions is Injective (1)
- Proposition: Composition of Bijective Functions is Bijective (1)
- Proposition: Injective, Surjective and Bijective Compositions (1)
- Proposition: The Inverse Of a Composition (1)

- 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)
- 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)
- Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice (1)
- 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)
- Proposition: Finite Chains are Well-ordered (1)

- 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)
- Definition: Contained Relation "`$\in_X$`"
- Proposition: Contained Relation is a Strict Order (1)
- Definition: Transitive Set (2)
- Definition: Extensional Relation (3)
- Explanation: Examples of Extensional Relations (related to Definition: Extensional Relation) (3)
- Proposition: The Contained Relation is Extensional (1)
- Proposition: Partial Orders are Extensional (1)
- Proposition: Strict Orders are Extensional (1)

- Explanation: Examples of Extensional Relations (related to Definition: Extensional Relation) (3)
- Definition: Well-founded Relation (1)
- Definition: Mostowski Function and Collapse (6)
- Example: Working with Mostowski Functions and Collapses (related to Definition: Mostowski Function and Collapse)
- Example: Some Other Examples Of Mostowski Functions and Collapses (related to Definition: Mostowski Function and Collapse)
- Motivation: Observation 1: The Mostowski Function Produces Transitive Sets (related to Definition: Mostowski Function and Collapse)
- Motivation: Observation 2: The Mostowski Function (Sometimes) Produces Relation Embeddings (related to Definition: Mostowski Function and Collapse)
- Theorem: Mostowski's Theorem (2)

- Definition: Ordinal Number (4)
- Proposition: Equivalent Notions of Ordinals (1)
- Proposition: Ordinals Are Downward Closed (1)
- Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals (1)
- Theorem: Trichotomy of Ordinals (1)

- Lemma: Properties of Ordinal Numbers (1)
- Lemma: Successor of Ordinal (1)
- 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)
- Definition: Finite Set, Infinite Set
- Definition: Comparison of Cardinal Numbers
- Chapter: Can Cardinals be Ordered?
- Theorem: Schröder-Bernstein Theorem (1)
- Chapter: Simple Facts Regarding Cardinals (5)
- Proposition: Subsets of Finite Sets (1)
- Theorem: Distinction Between Finite and Infinite Sets Using Subsets (1)
- Proposition: More Characterizations of Finite Sets (1)
- Lemma: Finite Cardinal Numbers and Set Operations (1)
- Proposition: Counting the Set's Elements Using Its Partition (1)

- 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)
- Proposition: Cardinals of a Set and Its Power Set (1)
- Proposition: Subset of a Countable Set is Countable (1)
- Proposition: Rational Numbers are Countable (1)
- Proposition: Real Numbers are Uncountable (2)
- Proposition: Uncountable and Countable Subsets of Natural Numbers (1)
- Chapter: Continuum Hypothesis

- Part: Solving Strategies and Sample Solutions to Problems in Set Theory (1)

- 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)
- Proposition: Addition of Natural Numbers Is Commutative (1)
- Proposition: Addition of Natural Numbers Is Cancellative (2)
- Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) (related to Proposition: Addition Of Natural Numbers) (1)
- Proposition: Uniqueness of Natural Zero (1)

- Definition: Multiplication of Natural Numbers (6)
- Proposition: Multiplication of Natural Numbers Is Associative (1)
- Proposition: Multiplication of Natural Numbers is Commutative (1)
- Proposition: Multiplication of Natural Numbers Is Cancellative (2)
- Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) (related to Definition: Multiplication of Natural Numbers) (1)
- Proposition: Uniqueness Of Natural One (1)

- Proposition: Uniqueness Of Predecessors Of Natural Numbers (1)
- Proposition: Inequality of Natural Numbers and Their Successors (1)

- Proposition: Addition Of Natural Numbers (7)
- Axiom: Peano Axioms (2)
- Proposition: Algebraic Structure Of Natural Numbers Together With Addition (1)
- Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication (1)
- Proposition: Distributivity Law For Natural Numbers (1)
- 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)
- Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities (1)
- Proposition: Order Relation for Natural Numbers, Revised (1)
- Proposition: Every Natural Number Is Greater or Equal Zero (1)
- Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation (1)
- Proposition: Comparing Natural Numbers Using the Concept of Addition (2)

- Proposition: Well-Ordering Principle of Natural Numbers (1)
- Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers (1)

- Definition: Set-theoretic Definitions of Natural Numbers (15)
- 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)
- Proposition: Addition of Integers Is Commutative (1)
- Proposition: Addition of Integers Is Cancellative (2)
- Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers) (1)
- Proposition: Uniqueness of Integer Zero (1)
- Proposition: Existence of Inverse Integers With Respect to Addition (1)

- Definition: Subtraction of Integers
- Proposition: Multiplication of Integers (8)
- Proof: (related to Proposition: Multiplication of Integers)
- Proposition: Multiplication of Integers Is Associative (1)
- Proposition: Multiplication of Integers Is Commutative (1)
- Proposition: Multiplication of Integers Is Cancellative (2)
- Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers) (1)
- Proposition: Uniqueness of Integer One (1)
- Proposition: Multiplying Negative and Positive Integers (1)

- Proposition: Distributivity Law For Integers (1)

- Proposition: Algebraic Structure of Integers Together with Addition (2)
- Proposition: Algebraic Structure of Integers Together with Addition and Multiplication (1)
- Definition: Order Relation for Integers - Positive and Negative Integers (2)
- Proposition: Order Relation for Integers is Strict Total (1)
- Definition: Absolute Value of Integers

- Proposition: Definition of Integers (20)
- 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)
- Proposition: Addition of Rational Numbers Is Commutative (1)
- Proposition: Addition of Rational Numbers Is Cancellative (2)
- Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers) (1)
- Proposition: Existence of Inverse Rational Numbers With Respect to Addition (2)
- Proposition: Uniqueness of Rational Zero (1)

- Proposition: Multiplication Of Rational Numbers (10)
- Proof: (related to Proposition: Multiplication Of Rational Numbers)
- Proposition: Multiplication of Rational Numbers Is Associative (1)
- Proposition: Multiplication Of Rational Numbers Is Commutative (1)
- Proposition: Multiplication Of Rational Numbers Is Cancellative (2)
- Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers) (1)
- Proposition: Uniqueness Of Rational One (1)
- Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication (1)
- Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication (1)
- Proposition: Multiplying Negative and Positive Rational Numbers (1)

- Definition: Subtraction of Rational Numbers
- Proposition: Distributivity Law For Rational Numbers (1)

- Proposition: Algebraic Structure of Rational Numbers Together with Addition (1)
- Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication (1)
- Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication (2)
- Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers (1)
- Definition: Absolute Value of Rational Numbers (1)

- Proposition: Definition of Rational Numbers (22)
- Part: Irrational Numbers (3)
- Definition: Definition of Irrational Numbers
- Proposition: Discovery of Irrational Numbers (2)

- Part: Real Numbers (73)
- Chapter: Real Numbers As Limits Of Rational Numbers (22)
- Definition: Rational Sequence
- Definition: Convergent Rational Sequence
- Definition: Rational Cauchy Sequence
- Proposition: Addition of Rational Cauchy Sequences (6)
- Proof: (related to Proposition: Addition of Rational Cauchy Sequences)
- Proposition: Addition of Rational Cauchy Sequences Is Associative (1)
- Proposition: Addition of Rational Cauchy Sequences Is Commutative (1)
- Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences) (1)
- Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition (1)
- Proposition: Addition of Rational Cauchy Sequences Is Cancellative (1)

- Proposition: Multiplication Of Rational Cauchy Sequences (5)
- Proof: (related to Proposition: Multiplication Of Rational Cauchy Sequences)
- Proposition: Multiplication of Rational Cauchy Sequences Is Associative (1)
- Proposition: Multiplication of Rational Cauchy Sequences Is Commutative (1)
- Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences) (1)
- Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative (1)

- Proposition: Distributivity Law For Rational Cauchy Sequences (1)
- Proposition: Rational Cauchy Sequence Members Are Bounded (1)
- Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition (1)
- Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication (1)
- Lemma: Unit Ring of All Rational Cauchy Sequences (1)
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are Rational Cauchy Sequences (1)
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are a Subgroup of Rational Cauchy Sequences With Respect To Addition (1)
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are an Ideal Of the Ring of Rational Cauchy Sequences (1)

- 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)
- Proposition: Addition Of Real Numbers Is Commutative (1)
- Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers) (1)
- Proposition: Existence of Inverse Real Numbers With Respect to Addition (1)
- Proposition: Uniqueness of Real Zero (1)
- 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)
- Corollary: `\((-x)y=-(xy)\)` (related to Proposition: Uniqueness of Negative Numbers) (2)
- Corollary: `\(-(-x)=x\)` (related to Proposition: Uniqueness of Negative Numbers) (1)

- Proposition: Addition of Real Numbers Is Cancellative (2)

- 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)
- Proposition: Multiplication of Real Numbers Is Commutative (2)
- Proposition: Multiplication of Real Numbers Is Cancellative (2)
- Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers) (1)
- Proposition: Uniqueness of Real One (1)
- Proposition: Existence of Inverse Real Numbers With Respect to Multiplication (1)
- 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)
- Corollary: `\((x^{-1})^{-1}=x\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication) (1)

- Proposition: Multiplying Negative and Positive Real Numbers (1)

- 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)

- Proposition: Unique Solvability of `\(a+x=b\)` (1)
- Proposition: Unique Solvability of `$ax=b$` (1)
- Proposition: `\(-(x+y)=-x-y\)` (1)
- Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)` (1)

- Proposition: Algebraic Structure of Real Numbers Together with Addition (1)
- Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication (1)
- Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication (2)
- Definition: Order Relation of Real Numbers (2)
- Axiom: Archimedean Axiom (5)
- Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle) (related to Axiom: Archimedean Axiom) (1)
- Corollary: Existence of Powers Exceeding Any Positive Constant (related to Axiom: Archimedean Axiom) (1)
- Corollary: Existence of Unique Integers Exceeding Real Numbers (related to Axiom: Archimedean Axiom) (1)
- Corollary: Existence of Arbitrarily Small Powers (related to Axiom: Archimedean Axiom) (1)
- Corollary: Existence of Arbitrarily Small Positive Rational Numbers (related to Axiom: Archimedean Axiom) (1)

- Definition: Absolute Value of Real Numbers (Modulus) (1)

- Chapter: Real Numbers As Limits Of Rational Numbers (22)
- Part: Complex Numbers (29)
- Chapter: Algebraic Properties of Complex Numbers (4)
- Proposition: Algebraic Structure of Complex Numbers Together with Addition (1)
- Proposition: Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication (1)
- Proposition: Algebraic Structure of Complex Numbers Together with Addition and Multiplication (1)
- Proposition: Complex Numbers are a Field Extension of Real Numbers (1)

- Definition: Definition of Complex Numbers (11)
- Definition: Addition of Complex Numbers (5)
- Proposition: Addition of Complex Numbers Is Associative (1)
- Proposition: Addition of Complex Numbers Is Commutative (1)
- Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers) (1)
- Proposition: Existence of Inverse Complex Numbers With Respect to Addition (1)
- Proposition: Uniqueness of Complex Zero (1)

- Definition: Subtraction of Complex Numbers
- Definition: Multiplication of Complex Numbers (4)
- Proposition: Multiplication of Complex Numbers Is Associative (1)
- Proposition: Multiplication of Complex Numbers Is Commutative (1)
- Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers) (1)
- Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication (1)

- Proposition: Distributivity Law for Complex Numbers (1)

- Definition: Addition of Complex Numbers (5)
- Chapter: Simple Formulas Involving the Complex Numbers (5)
- Proposition: Extracting the Real and the Imaginary Part of a Complex Number (1)
- Proposition: Calculating with Complex Conjugates (1)
- Proposition: Complex Numbers Cannot Be Ordered (1)
- Proposition: Absolute Value of Complex Conjugate (1)
- Proposition: Absolute Value of the Product of Complex Numbers (1)

- Chapter: Vector Properties of Complex Numbers (9)
- Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)` (1)
- Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis (1)
- Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers (1)
- 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)
- Proposition: Multiplication of Complex Numbers Using Polar Coordinates (1)

- Chapter: Algebraic Properties of Complex Numbers (4)
- 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)
- Definition: Euler's Constant
- Definition: Number `$\pi$`

- Chapter: Sum Manipulation Methods (22)
- Definition: Sums
- Proposition: Basic Rules of Manipulating Finite Sums (1)
- Proposition: Double Summation (1)
- Proposition: Rule of Combining Different Sets of Indices (1)
- Proposition: The General Perturbation Method (1)
- Proposition: Abelian Partial Summation Method (1)
- Section: Closed Formulas for Sums (15)
- Proposition: Sum of Consecutive Natural Numbers (1)
- Proposition: Sum of Consecutive Odd Numbers (1)
- Proposition: Sum of Squares (1)
- Proposition: Sum of Cube Numbers (1)
- Proposition: Geometric Sum (1)
- Proposition: Sum of Geometric Progression (1)
- Proposition: Sum of Arithmetic Progression (1)
- Proposition: Alternating Sum of Binomial Coefficients (1)
- Proposition: Sum of Cosines (1)
- Proposition: Sum of Binomial Coefficients (1)
- Proposition: Sum of Binomial Coefficients I (1)
- Proposition: Sum of Binomial Coefficients II (1)
- Proposition: Sum of Binomial Coefficients III (1)
- Proposition: Sum of Binomial Coefficients IV (1)
- Proposition: Sum of Factorials (I) (1)

- Proposition: Product of Two Sums (Generalized Distributivity Rule) (1)

- Chapter: Fractional Arithmetic (5)
- Definition: Ratio of Two Real Numbers
- Proposition: Equality of Two Ratios (1)
- Proposition: Sum and Difference of Two Ratios (1)
- Proposition: Product of Two Ratios (1)
- Proposition: Ratio of Two Ratios (1)

- Chapter: Sequences of Numbers (3)
- Section: Fibonacci Numbers
- Section: Fermat Numbers
- Definition: Triangle Numbers

- Chapter: Product Manipulation Methods (1)

- Part: Natural Numbers (30)
- 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)

- 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: Substructure (1)
- Definition: Algebraic Structure (Algebra)
- Definition: Binary Operation
- Chapter: Important Properties of Binary Operations (8)
- Definition: Closure
- Definition: Associativity (1)
- Definition: Commutativity (1)
- Definition: Existence of a Neutral Element
- Proposition: Uniqueness of the Neutral Element (1)
- Definition: Inverse Element
- Proposition: Uniqueness of Inverse Elements (1)
- 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)
- Definition: Exponentiation in a Group (1)
- 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)
- Explanation: Unions of Subgroups Are Not Subgroups (related to Chapter: Groups (Overview))
- Proposition: Unique Solvability of `$a\ast x=b$` in Groups (1)

- Motivation: Calculations in a Group (related to Chapter: Groups (Overview)) (3)
- 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

- Definition: Integral Element (1)

- Definition: Algebra over a Ring (2)
- 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)

- Chapter: Vector Spaces (Overview) (3)
- Definition: Vector Space (2)
- Definition: Subspace (1)

- Chapter: Modules (Overview) (1)
- Definition: Module

- Motivation: Common Concepts Of Algebra: Substructures and Morphisms (related to Part: Algebraic Structures - Overview) (8)
- 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)
- Example: Examples of Properties of Group Homomorphisms (related to Definition: Group Homomorphism)
- Lemma: Kernel and Image of Group Homomorphism (1)
- 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)

- Proposition: Additive Subgroups of Integers (1)
- Theorem: Construction of Groups from Commutative and Cancellative Semigroups (1)
- 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)
- Proposition: Group Homomorphisms with Cyclic Groups (1)
- Lemma: Cyclic Groups are Abelian (1)
- Lemma: Subgroups of Cyclic Groups (1)
- Proposition: Subgroups of Finite Cyclic Groups (1)
- Definition: Direct Product of Groups
- Definition: Conjugate Elements of a Group
- Definition: Cosets
- Proposition: Properties of Cosets (1)
- Lemma: Subgroups and Their Cosets are Equipotent (1)
- Theorem: Order of Subgroup Divides Order of Finite Group (1)
- Theorem: Order of Cyclic Group (Fermat's Little Theorem) (1)
- Chapter: Symmetry Groups
- Definition: Normal Subgroups
- Lemma: Factor Groups (1)
- Lemma: Group Homomorphisms and Normal Subgroups (1)
- Theorem: First Isomorphism Theorem for Groups (1)
- 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)
- 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)
- Definition: Zero Ring
- Theorem: Construction of Fields from Integral Domains (1)
- Theorem: Finite Integral Domains are Fields (1)

- Proposition: Generalization of Cancellative Multiplication of Integers (1)
- 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)

- Definition: Associate (1)
- 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)
- Definition: Divisibility of Ideals
- Definition: Addition of Ideals

- Proposition: Principal Ideals being Prime Ideals (1)
- Proposition: Principal Ideals being Maximal Ideals (1)
- Definition: Principal Ideal (5)
- Proposition: Principal Ideal Generated by A Unit (1)
- Proposition: Criterions for Equality of Principal Ideals (1)
- Lemma: Divisibility of Principal Ideals (1)
- 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)
- Proposition: Open and Closed Subsets of a Zariski Topology (1)
- Definition: Zariski Topology of a Commutative Ring

- Lemma: Fiber of Prime Ideals Under a Spectrum Function (2)

- Definition: Spectrum of a Commutative Ring (3)
- Definition: Maximal Ideal (1)
- Lemma: Fiber of Maximal Ideals (1)

- Lemma: Prime Ideals of Multiplicative Systems in Integral Domains (1)
- Theorem: Connection between Rings, Ideals, and Fields (1)

- Definition: Ideal (3)
- Lemma: Factor Rings, Generalization of Congruence Classes (2)

- Definition: Zero Divisor and Integral Domain (4)
- Chapter: Polynomial Rings, Irreducibility, and Field Extensions (11)
- Definition: Polynomial over a Ring, Degree, Variable (1)
- 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)

- Chapter: Algebraic and Transcendent Numbers (3)
- Theorem: Isomorphism of Rings
- Definition: Characteristic of a Ring (2)

- Chapter: Divisibility in General Rings (34)
- 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)

- 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)
- 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)
- 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)
- Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations (1)

- 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)
- Definition: Generating Systems
- Definition: Basis, Coordinate System (2)
- Lemma: Uniqueness Lemma of a Finite Basis (1)
- Theorem: Finite Basis Theorem (1)

- Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
- Definition: Linear Span
- Section: Cross Product
- Section: Subspaces (2)
- 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)
- Definition: Dot Product, Inner Product, Scalar Product (Complex Case) (1)

- 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)
- 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)

- Chapter: Introduction to Matrices and Vectors (13)
- 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)
- Proposition: Square of a Non-Zero Element is Positive in Ordered Fields (1)
- 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)
- Definition: Dependent and Independent Absolute Values
- Proposition: Characterization of Dependent Absolute Values (1)
- Proposition: Characterization of Non-Archimedean Absolute Values (1)
- 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)
- Problem: Verifying Subgroup Properties (1)

- Chapter: Group-theoretic Problems (2)

- Part: Algebraic Structures - Overview (54)
- 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)
- Definition: Nested Real Intervals
- Proposition: Limit of Nested Real Intervals (1)

- Proposition: Rational Numbers are Dense in Real Numbers (1)
- Proposition: Open Intervals Contain Uncountably Many Irrational Numbers (1)
- Proposition: Open Real Intervals are Uncountable (1)

- Section: Real Intervals and Bounded Real Sets (11)
- 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)
- Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty (1)
- Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity (1)

- Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
- Definition: Monotonic Sequences (1)
- Section: Theorems Regarding Limits Of Sequences (8)
- Proposition: Uniqueness Of the Limit of a Sequence (1)
- Proposition: Sum of Convergent Real Sequences (1)
- Proposition: Difference of Convergent Real Sequences (1)
- Proposition: Product of Convegent Real Sequences (1)
- Proposition: Product of a Real Number and a Convergent Real Sequence (1)
- Proposition: Quotient of Convergent Real Sequences (1)
- Proposition: How Convergence Preserves the Order Relation of Sequence Members (1)
- Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members (1)

- Definition: Limits of Real Functions
- Section: Theorems Regarding Limits of Functions (8)
- Proposition: Limit of a Function is Unique If It Exists (1)
- Proposition: Arithmetic of Functions with Limits - Sums (1)
- Proposition: Arithmetic of Functions with Limits - Difference (1)
- Proposition: Arithmetic of Functions with Limits - Division (1)
- Proposition: Preservation of Inequalities for Limits of Functions (1)
- Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity (1)
- Theorem: Squeezing Theorem for Functions (1)
- Proposition: Arithmetic of Functions with Limits - Product (1)

- Section: Examples of Limit Calculations (17)
- Proposition: Limit of 1/n (1)
- Proposition: Limit of the Constant Function (1)
- Proposition: Limit of the Identity Function (1)
- Proposition: Limit of Nth Powers (1)
- Proposition: Convergence Behavior of the Sequence `\((b^n)\)` (1)
- Proposition: Limit of a Polynomial (1)
- Proposition: Limit of a Rational Function (1)
- Proposition: Limit of Nth Root of a Positive Constant (1)
- Proposition: Limit of Nth Root of N (1)
- Proposition: Limits of General Powers (2)
- Proposition: Square Roots (1)
- Proposition: Limit of Exponential Growth as Compared to Polynomial Growth (1)
- Proposition: Limits of Logarithm in `$[0,+\infty]$` (1)
- Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth (1)
- Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function (1)
- Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function (1)

- Definition: Real Subsequence
- Theorem: Every Bounded Real Sequence has a Convergent Subsequence (1)
- 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)
- Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence (1)
- Definition: Isolated Point (Real Numbers)

- Definition: Asymptotical Approximation
- Lemma: Decreasing Sequence of Suprema of Extended Real Numbers (1)
- Definition: Limit Superior
- Lemma: Increasing Sequence of Infima of Extended Real Numbers (1)
- Definition: Limit Inferior

- Chapter: Completeness of Real Numbers (7)
- Proposition: The distance of real numbers makes real numbers a metric space. (1)
- Definition: Real Cauchy Sequence
- Theorem: Completeness Principle for Real Numbers (2)
- Theorem: Supremum Property, Infimum Property (1)
- Proposition: Convergent Real Sequences Are Cauchy Sequences (1)
- Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers (1)

- Chapter: Criteria for Convergence of Sequences (4)
- Theorem: Every Bounded Monotonic Sequence Is Convergent (2)
- Proposition: Convergent Real Sequences are Bounded (1)
- Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence (1)

- Theorem: Defining Properties of the Field of Real Numbers (1)
- Chapter: Useful Inequalities (21)
- Proposition: Generalized Triangle Inequality (1)
- Theorem: Triangle Inequality (1)
- Theorem: Reverse Triangle Inequalities (1)
- Definition: (Weighted) Arithmetic Mean
- Theorem: Inequality of the Arithmetic Mean (1)
- Theorem: Inequality of Weighted Arithmetic Mean (1)
- Theorem: Bernoulli's Inequality (1)
- Proposition: Generalized Bernoulli's Inequality (1)
- Proposition: Cauchy–Schwarz Inequality (1)
- Definition: Geometric Mean
- Theorem: Inequality Between the Geometric and the Arithmetic Mean (1)
- Lemma: Upper Bound for the Product of General Powers (1)
- Proposition: Hölder's Inequality (1)
- Proposition: Minkowski's Inequality (1)
- Proposition: Hölder's Inequality for Integral p-norms (1)
- Proposition: Cauchy-Schwarz Inequality for Integral p-norms (1)
- Proposition: Minkowski's Inequality for Integral p-norms (1)
- Proposition: Inequality between Square Numbers and Powers of `$2$` (1)
- Proposition: Inequality between Powers of `$2$` and Factorials (1)
- Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials (1)
- Proposition: Bounds for Partial Sums of Exponential Series (1)

- 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)
- Proposition: Composition of Continuous Functions at a Single Point (1)
- Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point (1)
- Theorem: Intermediate Value Theorem (1)
- Theorem: Intermediate Root Value Theorem (1)
- 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)
- 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)

- Definition: Continuous Real Functions (4)
- Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions (1)
- Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain (1)
- Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals (1)
- Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth (1)

- Definition: Even and Odd Complex Functions
- Section: Differentiable Functions (15)
- Definition: Difference Quotient
- Definition: Derivative, Differentiable Functions (1)
- Definition: Higher-Order Derivatives (1)
- Proposition: Generalized Product Rule (1)

- Definition: Local Extremum (3)
- Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum (2)
- Proposition: Sufficient Condition for a Local Extremum (1)

- Theorem: Rolle's Theorem (1)
- Proposition: Differentiable Functions and Tangent-Linear Approximation (2)
- Proposition: Characterization of Monotonic Functions via Derivatives (1)
- 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)

- Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule (1)
- Proposition: Chain Rule (1)

- Definition: Continuously Differentiable Functions
- Definition: Uniformly Continuous Functions (Real Case) (5)
- Proposition: Not all Continuous Functions are also Uniformly Continuous (1)
- Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions (1)
- Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous (2)
- Corollary: All Uniformly Continuous Functions are Continuous (related to Definition: Uniformly Continuous Functions (Real Case)) (1)

- Definition: Periodic Functions
- Definition: Even and Odd Functions (1)
- Proposition: Derivatives of Even and Odd Functions (1)

- Section: Integrable Functions (24)
- Subsection: Riemann Integral (22)
- Proposition: Riemann Integral for Step Functions (2)
- Definition: Riemann-Integrable Functions (8)
- Proposition: Step Function on Closed Intervals are Riemann-Integrable (1)
- Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable (1)
- Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable (1)
- Proposition: Linearity and Monotony of the Riemann Integral (1)
- Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable (1)
- Proposition: Product of Riemann-integrable Functions is Riemann-integrable (1)
- 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)

- Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions (2)
- Definition: Riemann Sum With Respect to a Partition (1)
- Theorem: Indefinite Integral, Antiderivative (2)
- Theorem: Fundamental Theorem of Calculus (1)
- Proposition: Integrals on Adjacent Intervals (1)
- Theorem: Integration by Substitution (1)
- Theorem: Mean Value Theorem For Riemann Integrals (1)
- Theorem: Partial Integration (1)
- Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine (1)
- Lemma: Trapezoid Rule (1)

- Subsection: Extended Concept of the Riemann Integral - the Improper Integral (1)
- Definition: Improper Integral

- Subsection: Riemann-Stieltjes Integral

- Subsection: Riemann Integral (22)
- Lemma: Invertible Functions on Real Intervals (2)
- Definition: Convex and Concave Functions (2)
- 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)
- Definition: Supremum Norm for Functions
- Proposition: Supremum Norm and Uniform Convergence (1)
- Proposition: Uniform Convergence Criterion of Cauchy (1)
- Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series (1)
- Proposition: Calculations with Uniformly Convergent Functions (1)

- 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)

- Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals (2)

- Definition: Continuous Functions at Single Real Numbers (9)
- Chapter: Types of Real Functions (86)
- Definition: Constant Function Real Case (1)
- Definition: Real Identity Function (1)
- Proposition: Identity Function is Continuous (1)

- Definition: Reciprocal Function (2)
- Proposition: Derivative of the Reciprocal Function (1)
- Proposition: Integral of the Reciprocal Function (1)

- Definition: Real Absolute Value Function (1)
- Definition: Step Functions (2)
- Definition: Positive and Negative Parts of a Real-Valued Function
- Definition: Linear Function (2)
- Proposition: Nth Roots of Positive Numbers (1)
- Proposition: Rational Powers of Positive Numbers (2)
- Proposition: Nth Powers (2)
- Proposition: General Powers of Positive Numbers (4)
- Proof: (related to Proposition: General Powers of Positive Numbers)
- Proposition: Calculation Rules for General Powers (1)
- Proposition: Derivative of General Powers of Positive Numbers (1)
- Proposition: Integral of General Powers (1)

- Definition: Polynomials (3)
- Proposition: Limits of Polynomials at Infinity (2)
- Proposition: Eveness (Oddness) of Polynomials (1)

- Definition: Rational Functions (1)
- Proposition: Rational Functions are Continuous (1)

- 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)
- Proposition: Functional Equation of the Exponential Function of General Base (2)
- Proposition: Exponential Function of General Base With Natural Exponents (1)
- Proposition: Exponential Function of General Base With Integer Exponents (1)
- Proposition: Functional Equation of the Exponential Function of General Base (Revised) (1)

- Proposition: Estimate for the Remainder Term of Exponential Function (1)
- 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)
- Corollary: Exponential Function Is Strictly Monotonically Increasing (related to Proposition: Functional Equation of the Exponential Function) (1)
- Corollary: Exponential Function Is Non-Negative (Real Case) (related to Proposition: Functional Equation of the Exponential Function) (1)
- Corollary: Exponential Function and the Euler Constant (related to Proposition: Functional Equation of the Exponential Function) (1)

- Proposition: Continuity of Exponential Function (1)
- Proposition: `\(\exp(0)=1\)` (1)
- Proposition: Derivative of the Exponential Function (1)
- Proposition: Integral of the Exponential Function (1)

- Section: Logarithms (5)
- Proposition: Natural Logarithm (4)
- Proof: (related to Proposition: Natural Logarithm)
- Proposition: Functional Equation of the Natural Logarithm (1)
- Proposition: Derivative of the Natural Logarithm (1)
- Proposition: Integral of the Natural Logarithm (1)

- Proposition: Logarithm to a General Base (1)

- Proposition: Natural Logarithm (4)
- Section: Trigonometric Functions (30)
- Definition: Cosine of a Real Variable (5)
- Proposition: Eveness of the Cosine of a Real Variable (1)
- Proposition: Zero of Cosine (1)
- Proposition: Derivative of Cosine (1)
- Proposition: Integral of Cosine (1)
- Corollary: Representing Real Cosine by Complex Exponential Function (related to Definition: Cosine of a Real Variable) (1)

- Definition: Sine of a Real Variable (4)
- Proposition: Oddness of the Sine of a Real Variable (1)
- Proposition: Derivative of Sine (1)
- Proposition: Integral of Sine (1)
- Corollary: Representing Real Sine by Complex Exponential Function (related to Definition: Sine of a Real Variable) (1)

- Definition: Tangent of a Real Variable (2)
- Proposition: Additivity Theorem of Tangent (1)
- Proposition: Derivative of Tangent (1)

- Proposition: Continuity of Cosine and Sine (1)
- Proposition: Pythagorean Identity (1)
- Proposition: Additivity Theorems of Cosine and Sine (1)
- Proposition: Infinite Series for Cosine and Sine (2)
- 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)
- Corollary: Negative Cosine and Sine vs Shifting the Argument (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
- 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)
- Corollary: All Zeros of Cosine and Sine (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)
- Corollary: More Insight to Euler's Identity (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function) (1)

- Proposition: Inverse Cosine of a Real Variable (1)
- 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)
- Proposition: Integral of Inverse Sine (1)

- 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)
- Proposition: Derivative of the Inverse Tangent (1)
- Proposition: Integral of the Inverse Tangent (1)

- Definition: Cosine of a Real Variable (5)
- Section: Cyclometric Functions
- Section: Hyporbolic Functions (3)
- Definition: Hyperbolic Cosine (1)
- Proposition: Sum of Arguments of Hyperbolic Cosine (1)

- Definition: Hyperbolic Sine (1)
- Proposition: Sum of Arguments of Hyperbolic Sine (1)

- Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine (1)

- Definition: Hyperbolic Cosine (1)
- Section: Inverse Hyperbolic Functions (2)
- Proposition: Inverse Hyperbolic Sine (1)
- Proposition: Inverse Hyperbolic Cosine (1)

- Section: Mixed Functions
- Proposition: Gamma Function (2)
- 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)
- Definition: Infinite Series, Partial Sums
- Definition: Convergent Real Series (4)
- Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent (1)
- Proposition: Sum of Convergent Real Series (1)
- Proposition: Difference of Convergent Real Series (1)
- Proposition: Product of a Real Number and a Convergent Real Series (1)

- Definition: Absolutely Convergent Series (2)
- Proposition: Convergence Behaviour of Absolutely Convergent Series (1)
- Proposition: Cauchy Product of Absolutely Convergent Series (1)

- Definition: Complex Infinite Series
- Definition: Absolutely Convergent Complex Series (3)
- Proposition: Direct Comparison Test For Absolutely Convergent Complex Series (1)
- Proposition: Ratio Test For Absolutely Convergent Complex Series (1)
- Proposition: Cauchy Product of Absolutely Convergent Complex Series (1)

- Definition: Divergent Series
- Section: Convergence and Divergence Criteria for Real Series (20)
- Proposition: Monotony Criterion (2)
- Proposition: Leibniz Criterion for Alternating Series (1)
- Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero (1)
- Proposition: Cauchy Condensation Criterion (2)
- Proposition: Direct Comparison Test For Absolutely Convergent Series (1)
- Proposition: Direct Comparison Test For Divergent Series (1)
- Proposition: Limit Comparizon Test (1)
- Proposition: Root Test (1)
- Proposition: Ratio Test (1)
- Proposition: Limit Test for Roots or Ratios (1)
- Lemma: Convergence Test for Telescoping Series (1)
- Proposition: Raabe's Test (1)
- Subsection: Convergence Criteria for Infinite Series Involving Products (4)
- Lemma: Abel's Lemma for Testing Convergence (1)
- Proposition: Cauchy-Schwarz Test (1)
- Proposition: Abel's Test (1)
- Proposition: Dirichlet's Test (1)

- Proposition: Integral Test for Convergence (1)
- Proposition: Cauchy Criterion (1)

- Definition: Rearrangement of Infinite Series (2)
- Section: Closed Formulas for Infinite Real Series (1)
- Proposition: Infinite Geometric Series (1)

- Definition: `\(b\)`-Adic Fractions (4)
- Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences (1)
- Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions (2)
- 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)
- Proposition: Taylor's Formula with Remainder Term of Lagrange (1)
- Theorem: Taylor's Formula (2)

- Chapter: Power Series Introduction
- Chapter: Fourier Series

- Chapter: Basics of Real Analysis of One Variable (14)
- 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)

- Chapter: Differentiability (6)
- 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)
- Proposition: Complex Convergent Sequences are Bounded (1)
- Definition: Continuous Functions at Single Complex Numbers
- Proposition: Convergent Complex Sequences Are Cauchy Sequences (1)
- Definition: Complex Sequence
- Definition: Bounded Complex Sequences (1)
- Definition: Convergent Complex Sequence (5)
- Proposition: Quotient of Convergent Complex Sequences (1)
- Proposition: Sum of Convergent Complex Sequences (1)
- Proposition: Difference of Convergent Complex Sequences (1)
- Proposition: Product of Convegent Complex Sequences (1)
- Proposition: Product of a Complex Number and a Convergent Complex Sequence (1)

- Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences (2)
- Definition: Limits of Complex Functions
- Definition: Complex Cauchy Sequence
- Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences (1)
- Theorem: Completeness Principle for Complex Numbers (1)
- 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

- Section: Moebius Transformations (3)
- 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)
- Proposition: Functional Equation of the Complex Exponential Function (2)
- Proposition: `\(\exp(0)=1\)` (Complex Case) (1)
- Proposition: Continuity of Complex Exponential Function (1)
- Proposition: Complex Conjugate of Complex Exponential Function (1)

- Lemma: Unit Circle (1)
- Proposition: Euler's Formula (1)
- Proposition: n-th Roots of Unity (1)
- Theorem: De Moivre's Identity, Complex Powers (1)
- 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)

- Proposition: Complex Exponential Function (7)
- 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)

- Chapter: Topological Aspects (25)
- Part: Differential Equations (14)
- Chapter: Classification of Differential Equations (5)
- Definition: First-Order Ordinary Differential Equation (ODE) (1)
- Section: Linear vs. Non-Linear DE
- Section: Order of DE
- Section: Systems of DE
- Proposition: Legendre Polynomials and Legendre Differential Equations (1)

- 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

- Chapter: Classification of Differential Equations (5)
- 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)
- 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)
- Proposition: Image of a Compact Set Under a Continuous Function (1)
- Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space (1)
- Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space (1)
- Proposition: Closed n-Dimensional Cuboids Are Compact (2)
- Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed (1)
- Proposition: Convergent Sequences are Bounded (1)
- Theorem: Heine-Borel Theorem (1)
- Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum (1)
- 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)

- Proposition: Closed Subsets of Compact Sets are Compact (1)

- 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)
- Definition: Discrete and Indiscrete Topology
- Definition: Ordering of Topologies
- Definition: Open, Closed, Clopen
- Proposition: Alternative Characterization of Topological Spaces (1)
- Definition: Neighborhood
- Proposition: Properties of the Set of All Neighborhoods of a Point (1)
- Proposition: A Necessary Condition of a Neighborhood to be Open (1)
- Definition: Boundary Points, Closures, Interiors, and Exteriors
- Proposition: How the Boundary Changes the Property of a Set of Being Open (1)
- 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)
- 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)
- Definition: Comparison of Filters, Finer and Coarser Filters
- Axiom: Filter
- Proposition: Filter Base (1)
- 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)
- Proposition: Continuity of Compositions of Functions (1)
- Definition: Open and Closed Functions
- Proposition: Bijective Open Functions (1)
- Definition: Homeomorphism, Homeomorphic Spaces
- Proposition: Equivalent Notions of Homeomorphisms (1)
- Definition: Topological, Continuous, Open, and Closed Invariants

- Definition: Limit of a Function
- Definition: Convergent Sequences and Limits (1)
- Proposition: Uniqueness of the Limit of a Sequence (1)
- Definition: Cauchy Sequence (1)
- Theorem: Theorem of Bolzano-Weierstrass (1)
- Definition: Norm, Normed Vector Space (3)
- Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm (2)
- Proposition: Integral p-Norm (1)

- Definition: Isometry (1)
- Proposition: Isometry is Injective (1)

- Definition: Bounded Sequence
- Definition: Bounded Subset of a Metric Space
- Proposition: Metric Spaces are Hausdorff Spaces (1)
- Proposition: Distance in Normed Vector Spaces (1)
- Definition: Open Cover
- Definition: Continuous Functions in Metric Spaces (6)
- Definition: Pointwise and Uniform Convergence (2)
- 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)
- Proposition: Modulus of Continuity is Monotonically Increasing (1)
- Proposition: Modulus of Continuity is Continuous (1)

- Definition: Uniformly Continuous Functions (General Metric Spaces Case) (1)

- Chapter: Basic Topological Concepts (13)
- Part: Separation Of Topological Spaces (5)
- Axiom: Separation Axioms
- Proposition: Characterization of `$T_1$` Spaces (1)
- Proposition: Inheritance of the `$T_1$` Property (1)
- Proposition: Characterization of `$T_2$` Spaces (1)
- Proposition: Inheritance of the `$T_2$` Property (1)

- Part: Metric Spaces (7)
- Definition: Metric (Distance) (1)
- Definition: Metric Space
- Definition: Open Sets in Metric Spaces (2)
- Definition: Open Ball, Neighborhood
- Definition: Diameter In Metric Spaces
- Proposition: Metric Spaces and Empty Sets are Clopen (1)

- 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

- Definition: Manifold (6)
- 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)
- Definition: 1.02: Line, Curve (1)
- 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)
- 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: 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)
- 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)
- Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment (1)
- Proposition: 1.03: Cutting a Segment at a Given Size (1)
- Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle (1)
- Proposition: 1.05: Isosceles Triangles I (2)
- Proposition: 1.06: Isosceles Triagles II (2)
- Proposition: 1.07: Uniqueness of Triangles (1)
- Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles (1)
- Proposition: 1.09: Bisecting an Angle (1)
- Proposition: 1.10: Bisecting a Segment (1)
- Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line (1)
- Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line (1)
- 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)
- Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines) (1)

- Proposition: 1.14: Combining Rays to Straight Lines (1)
- Proposition: 1.15: Opposite Angles on Intersecting Straight Lines (1)
- Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles (2)
- Proposition: 1.17: The Sum of Two Angles of a Triangle (1)
- Proposition: 1.18: Angles and Sides in a Triangle I (1)
- Proposition: 1.19: Angles and Sides in a Triangle II (1)
- Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality) (1)
- Proposition: 1.21: Triangles within Triangles (1)
- Proposition: 1.22: Construction of Triangles From Arbitrary Segments (1)
- Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle (1)
- Proposition: 1.24: Angles and Sides in a Triangle III (1)
- Proposition: 1.25: Angles and Sides in a Triangle IV (2)
- Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles (1)
- Proposition: 1.27: Parallel Lines I (1)
- Proposition: 1.28: Parallel Lines II (1)
- Proposition: 1.29: Parallel Lines III (2)
- Proposition: 1.30: Transitivity of Parallel Lines (1)
- Proposition: 1.31: Constructing a Parallel Line from a Line and a Point (1)
- 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)
- Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
- Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)
- 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)
- Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle) (1)

- Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram (1)
- 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)
- Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
- 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)
- Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
- Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)
- Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms) (1)

- Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels (1)
- Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels (1)
- Proposition: 1.37: Triangles of Equal Area I (1)
- Proposition: 1.38: Triangles of Equal Area II (1)
- Proposition: 1.39: Triangles of Equal Area III (1)
- Proposition: 1.40: Triangles of Equal Area IV (1)
- Proposition: 1.41: Parallelograms and Triagles (1)
- Proposition: 1.42: Construction of Parallelograms I (1)
- Proposition: 1.43: Complementary Segments of Parallelograms (1)
- Proposition: 1.44: Construction of Parallelograms II (1)
- Proposition: 1.45: Construction of Parallelograms III (1)
- Proposition: 1.46: Construction of a Square on a Given Segment (2)
- Proposition: 1.47: Pythagorean Theorem (1)
- Proposition: 1.48: The Converse of the Pythagorean Theorem (1)

- Subsection: Definitions from Book 1 (28)
- 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

- Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides (1)
- Subsection: Propositions from Book 2 (14)
- Proposition: 2.01: Summing Areas or Rectangles (1)
- Proposition: 2.02: Square is Sum of Two Rectangles (1)
- Proposition: 2.03: Rectangle is Sum of Square and Rectangle (1)
- Proposition: 2.04: Square of Sum (1)
- Proposition: 2.05: Rectangle is Difference of Two Squares (1)
- Proposition: 2.06: Square of Sum with One Halved Summand (1)
- Proposition: 2.07: Sum of Squares (1)
- Proposition: 2.08: Square of Sum with One Doubled Summand (1)
- Proposition: 2.09: Sum of Squares of Sum and Difference (1)
- Proposition: 2.10: Sum of Squares (Half) (1)
- Proposition: 2.11: Constructing the Golden Ratio of a Segment (1)
- Proposition: 2.12: Law of Cosines (for Obtuse Angles) (1)
- Proposition: 2.13: Law of Cosines (for Acute Angles) (1)
- Proposition: 2.14: Constructing a Square from a Rectilinear Figure (1)

- Subsection: Definitions from Book 2 (2)
- 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)
- Proposition: 3.02: Chord Lies Inside its Circle (1)
- Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector (1)
- Proposition: 3.04: Chords do not Bisect Each Other (1)
- Proposition: 3.05: Intersecting Circles have Different Centers (1)
- Proposition: 3.06: Touching Circles have Different Centers (1)
- Proposition: 3.07: Relative Lengths of Lines Inside Circle (1)
- Proposition: 3.08: Relative Lengths of Lines Outside Circle (1)
- Proposition: 3.09: Condition for Point to be Center of Circle (1)
- Proposition: 3.10: Two Circles have at most Two Points of Intersection (1)
- Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally (1)
- Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally (1)
- Proposition: 3.13: Circles Touch at One Point at Most (1)
- Proposition: 3.14: Equal Chords in Circle (1)
- Proposition: 3.15: Relative Lengths of Chords of Circles (1)
- 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)

- Proposition: 3.17: Construction of Tangent from Point to Circle (1)
- Proposition: 3.18: Radius at Right Angle to Tangent (1)
- Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center (1)
- Proposition: 3.20: Inscribed Angle Theorem (1)
- Proposition: 3.21: Angles in Same Segment of Circle are Equal (1)
- Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral (1)
- Proposition: 3.23: Segment on Given Base Unique (1)
- Proposition: 3.24: Similar Segments on Equal Bases are Equal (1)
- Proposition: 3.25: Construction of Circle from Segment (1)
- Proposition: 3.26: Equal Angles and Arcs in Equal Circles (1)
- Proposition: 3.27: Angles on Equal Arcs are Equal (1)
- Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles (1)
- Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines (1)
- Proposition: 3.30: Bisection of Arc (1)
- Proposition: 3.31: Relative Sizes of Angles in Segments (1)
- Proposition: 3.32: Angles made by Chord with Tangent (1)
- Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle (1)
- Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle (1)
- Proposition: 3.35: Intersecting Chord Theorem (1)
- Proposition: 3.36: Tangent Secant Theorem (1)
- Proposition: 3.37: Converse of Tangent Secant Theorem (1)

- Subsection: Definitions from Book 3 (11)
- 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)
- Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles (1)
- Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles (1)
- Proposition: 4.04: Inscribing Circle in Triangle (1)
- Proposition: 4.05: Circumscribing Circle about Triangle (1)
- Proposition: 4.06: Inscribing Square in Circle (1)
- Proposition: 4.07: Circumscribing Square about Circle (1)
- Proposition: 4.08: Inscribing Circle in Square (1)
- Proposition: 4.09: Circumscribing Circle about Square (1)
- Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex (1)
- Proposition: 4.11: Inscribing Regular Pentagon in Circle (1)
- Proposition: 4.12: Circumscribing Regular Pentagon about Circle (1)
- Proposition: 4.13: Inscribing Circle in Regular Pentagon (1)
- Proposition: 4.14: Circumscribing Circle about Regular Pentagon (1)
- 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)

- Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle (1)

- Subsection: Definitions from Book 4 (7)
- 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)
- Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition (1)
- Proposition: 5.03: Multiplication of Numbers is Associative (1)
- Proposition: 5.04: Multiples of Terms in Equal Ratios (1)
- Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction (1)
- Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction (1)
- Proposition: 5.07: Ratios of Equal Magnitudes (2)
- Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes (1)
- Proposition: 5.09: Magnitudes with Same Ratios are Equal (1)
- Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios (1)
- Proposition: 5.11: Equality of Ratios is Transitive (1)
- Proposition: 5.12: Sum of Components of Equal Ratios (1)
- Proposition: 5.13: Relative Sizes of Proportional Magnitudes (1)
- Proposition: 5.14: Relative Sizes of Components of Ratios (1)
- Proposition: 5.15: Ratio Equals its Multiples (1)
- Proposition: 5.16: Proportional Magnitudes are Proportional Alternately (1)
- Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated (1)
- Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded (1)
- Proposition: 5.19: Proportional Magnitudes have Proportional Remainders (2)
- Proposition: 5.20: Relative Sizes of Successive Ratios (1)
- Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion (1)
- Proposition: 5.22: Equality of Ratios Ex Aequali (1)
- Proposition: 5.23: Equality of Ratios in Perturbed Proportion (1)
- Proposition: 5.24: Sum of Antecedents of Proportion (1)
- Proposition: 5.25: Sum of Antecedent and Consequent of Proportion (1)

- Subsection: Definitions from Book 5 (18)
- 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)
- Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally (1)
- Proposition: 6.03: Angle Bisector Theorem (1)
- Proposition: 6.04: Equiangular Triangles are Similar (1)
- Proposition: 6.05: Triangles with Proportional Sides are Similar (1)
- Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar (1)
- Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar (1)
- Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles (2)
- Proposition: 6.09: Construction of Part of Line (1)
- Proposition: 6.10: Construction of Similarly Cut Straight Line (1)
- Proposition: 6.11: Construction of Segment in Squared Ratio (1)
- Proposition: 6.12: Construction of Fourth Proportional Straight Line (1)
- Proposition: 6.13: Construction of Mean Proportional (1)
- Proposition: 6.14: Characterization of Congruent Parallelograms (1)
- Proposition: 6.15: Characterization of Congruent Triangles (1)
- Proposition: 6.16: Rectangles Contained by Proportional Straight Lines (1)
- Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines (1)
- Proposition: 6.18: Construction of Similar Polygon (1)
- Proposition: 6.19: Ratio of Areas of Similar Triangles (2)
- Proposition: 6.20: Similar Polygons are Composed of Similar Triangles (2)
- Proposition: 6.21: Similarity of Polygons is Transitive (1)
- Proposition: 6.22: Similar Figures on Proportional Straight Lines (1)
- Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms (1)
- Proposition: 6.24: Parallelograms About Diameter are Similar (1)
- Proposition: 6.25: Construction of Figure Similar to One and Equal to Another (1)
- Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter (1)
- Proposition: 6.27: Similar Parallelogram on Half a Straight Line (1)
- Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram (1)
- Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram (1)
- Proposition: 6.30: Construction of the Inverse Golden Section (1)
- Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle (1)
- Proposition: 6.32: Triangles with Two Sides Parallel and Equal (1)
- Proposition: 6.33: Angles in Circles have Same Ratio as Arcs (1)

- Subsection: Definitions from Book 6 (3)
- 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)
- Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm (2)
- Proposition: 7.03: Greatest Common Divisor of Three Numbers (1)
- Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers (1)
- Proposition: 7.05: Divisors Obey Distributive Law (Sum) (1)
- Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum) (1)
- Proposition: 7.07: Divisors Obey Distributive Law (Difference) (1)
- Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference) (1)
- Proposition: 7.09: Alternate Ratios of Equal Fractions (1)
- Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions (1)
- Proposition: 7.11: Proportional Numbers have Proportional Differences (1)
- Proposition: 7.12: Ratios of Numbers is Distributive over Addition (1)
- Proposition: 7.13: Proportional Numbers are Proportional Alternately (1)
- Proposition: 7.14: Proportion of Numbers is Transitive (1)
- Proposition: 7.15: Alternate Ratios of Multiples (1)
- Proposition: 7.16: Natural Number Multiplication is Commutative (1)
- Proposition: 7.17: Multiples of Ratios of Numbers (1)
- Proposition: 7.18: Ratios of Multiples of Numbers (1)
- Proposition: 7.19: Relation of Ratios to Products (1)
- Proposition: 7.20: Ratios of Fractions in Lowest Terms (1)
- Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms (1)
- Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime (1)
- Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other (1)
- Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole (1)
- Proposition: 7.25: Square of Co-prime Number is Co-prime (1)
- Proposition: 7.26: Product of Co-prime Pairs is Co-prime (1)
- Proposition: 7.27: Powers of Co-prime Numbers are Co-prime (1)
- Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both (1)
- Proposition: 7.29: Prime not Divisor implies Co-prime (1)
- Proposition: 7.30: Euclidean Lemma (1)
- Proposition: 7.31: Existence of Prime Divisors (1)
- Proposition: 7.32: Natural Number is Prime or has Prime Factor (1)
- Proposition: 7.33: Least Ratio of Numbers (1)
- Proposition: 7.34: Existence of Least Common Multiple (1)
- Proposition: 7.35: Least Common Multiple Divides Common Multiple (1)
- Proposition: 7.36: Least Common Multiple of Three Numbers (1)
- Proposition: 7.37: Integer Divided by Divisor is Integer (1)
- Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer (1)
- Proposition: 7.39: Least Number with Three Given Fractions (1)

- Subsection: Definitions from Book 7 (22)
- 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)
- Proposition: 8.02: Construction of Geometric Progression in Lowest Terms (2)
- Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes (1)
- Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios (1)
- Proposition: 8.05: Ratio of Products of Sides of Plane Numbers (1)
- Proposition: 8.06: First Element of Geometric Progression not dividing Second (1)
- Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second (1)
- Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements (1)
- Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers (1)
- Proposition: Prop. 8.10: Product of Geometric Progressions from One (1)
- Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional (1)
- Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals (1)
- Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression (1)
- Proposition: Prop. 8.14: Number divides Number iff Square divides Square (1)
- Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube (1)
- Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square (1)
- Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube (1)
- Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional (1)
- Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals (1)
- Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane (1)
- Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid (1)
- Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square (1)
- Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube (1)
- Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square (1)
- Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube (1)
- Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares (1)
- Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes (1)
- Proof:

- Subsection: Propositions from Book 8 (28)

- Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines (106)

- Chapter: Euclid's “Elements” (670)