Branches History Index

Index: By Building Blocks

Mathematical Language

Axioms (34)
- Algebra (5)
  - Axiom: Axiom of Distributivity
  - Axiom: Axioms of Group
  - Axiom: Axioms of Magma
  - Axiom: Axioms of Monoid
  - Axiom: Axioms of Semigroup
- Geometry (7)
  - Euclidean Geometry (7)
    - Axiom: "Between" Relation, Axioms of Order
    - Axiom: Axioms of Connection
    - Elements Euclid (5)
      - Book 1 Plane Geometry (5)
        
        Axiom: 1.1: Straight Line Determined by Two Distinct Points
        
        Axiom: 1.2: Segment Extension
        
        Axiom: 1.3: Circle Determined by its Center and its Radius
        
        Axiom: 1.4: Equality of all Right Angles
        
        Axiom: 1.5: Parallel Postulate
- Logic (1)
  - Axiom: Bivalence of Truth
- Number Systems Arithmetics (2)
  - Axiom: Archimedean Axiom
  - Axiom: Peano Axioms
- Probability Theory And Statistics (3)
  - Axiom: Addition of the Probability of Mutually Exclusive Events
  - Axiom: Probability as a Non-Negative Number
  - Axiom: Probability of the Certain Event
- Set Theory (12)
  - Axiom: Axiom of Choice
  - Axiom: Axiom of Empty Set
  - Axiom: Axiom of Existence
  - Axiom: Axiom of Extensionality
  - Axiom: Axiom of Foundation
  - Axiom: Axiom of Infinity
  - Axiom: Axiom of Pairing
  - Axiom: Axiom of Power Set
  - Axiom: Axiom of Replacement (Schema)
  - Axiom: Axiom of Union
  - Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)
  - Axiom: Zermelo-Fraenkel Axioms
- Theoretical Physics (2)
  - Special Relativity (2)
    - Axiom: Principle of Relativity
    - Axiom: Principle of the Constancy of Light Speed
- Topology (2)
  - Axiom: Filter
  - Axiom: Separation Axioms
Definitions (759)
- Algebra (138)
  - Definition: (Unit) Ring
  - Definition: Absolute Values and Non-Archimedean Absolute Values of Fields
  - Definition: Addition of Ideals
  - Definition: Affine Basis, Affine Coordinate System
  - Definition: Affine Space
  - Definition: Affine Subspace
  - Definition: Affinely Dependent and Affinely Independent Points
  - Definition: Algebra over a Ring
  - Definition: Algebraic Element
  - Definition: Algebraic Structure (Algebra)
  - Definition: Alternating Multilinear Map
  - Definition: Associate
  - Definition: Associativity
  - Definition: Automorphism
  - Definition: Basis, Coordinate System
  - Definition: Bilinear Form
  - Definition: Binary Operation
  - Definition: Bounded Affine Set
  - Definition: Cancellation Property
  - Definition: Characteristic of a Field
  - Definition: Characteristic of a Ring
  - Definition: Closure
  - Definition: Coefficient Matrix
  - Definition: Column Vectors and Row Vectors
  - Definition: Commutative (Abelian) Group
  - Definition: Commutative (Unit) Ring
  - Definition: Commutativity
  - Definition: Complete Ordered Field
  - Definition: Conjugate Elements of a Group
  - Definition: Continued Fractions
  - Definition: Convex Affine Set
  - Definition: Convex Hull
  - Definition: Cosets
  - Definition: Cyclic Group, Order of an Element
  - Definition: Dependent and Independent Absolute Values
  - Definition: Diagonal Matrix
  - Definition: Dimension of a Vector Space
  - Definition: Dimension of an Affine Space
  - Definition: Direct Product of Groups
  - Definition: Direct Sum of Vector Spaces
  - Definition: Divisibility of Ideals
  - Definition: Dot Product, Inner Product, Scalar Product (Complex Case)
  - Definition: Dot Product, Inner Product, Scalar Product (General Field Case)
  - Definition: Eigenvalue
  - Definition: Eigenvector
  - Definition: Elementary Gaussian Operations
  - Definition: Elementary Symmetric Functions
  - Definition: Endomorphism
  - Definition: Epimorphism
  - Definition: Euclidean Affine Space
  - Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder
  - Definition: Existence of a Neutral Element
  - Definition: Exponentiation in a Group
  - Definition: Exponentiation in a Monoid
  - Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
  - Definition: Factorial Ring, Generalization of Factorization
  - Definition: Field
  - Definition: Field Extension
  - Definition: Field Homomorphism
  - Definition: Finite Field
  - Definition: Finite Field Extension
  - Definition: Gaussian Method to Solve Systems of Linear Equations, Rank of a Matrix
  - Definition: Generalization of Divisor and Multiple
  - Definition: Generalization of the Greatest Common Divisor
  - Definition: Generalization of the Least Common Multiple
  - Definition: Generating Set of a Group
  - Definition: Generating Set of an Ideal
  - Definition: Generating Systems
  - Definition: Group
  - Definition: Group Homomorphism
  - Definition: Group Operation
  - Definition: Group Order
  - Definition: Homomorphism
  - Definition: Ideal
  - Definition: Identity Matrix
  - Definition: Integral Closure
  - Definition: Integral Element
  - Definition: Inverse Element
  - Definition: Invertible and Inverse Matrix
  - Definition: Irreducible Polynomial
  - Definition: Irreducible, Prime
  - Definition: Isomorphism
  - Definition: Linear Combination
  - Definition: Linear Equations with many Unknowns
  - Definition: Linear Map
  - Definition: Linear Span
  - Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector
  - Definition: Magma
  - Definition: Matrix Multiplication
  - Definition: Matrix and Vector Addition
  - Definition: Matrix, Set of Matrices over a Field
  - Definition: Maximal Ideal
  - Definition: Minimal Polynomial
  - Definition: Module
  - Definition: Monoid
  - Definition: Monomorphism
  - Definition: Multilinear Map
  - Definition: Multiplicative System
  - Definition: Multiplicity of a Root of a Polynomial
  - Definition: Normal Subgroups
  - Definition: Ordered Field
  - Definition: Points, Lines, Planes, Hyperplanes
  - Definition: Polynomial Ring
  - Definition: Polynomial over a Ring, Degree, Variable
  - Definition: Prime Field
  - Definition: Prime Ideal
  - Definition: Principal Ideal
  - Definition: Principal Ideal Domain
  - Definition: Principal Ideal Ring
  - Definition: Recursive Definition of the Determinant
  - Definition: Reduction of an Integer Polynomial Modulo a Prime Number
  - Definition: Ring Homomorphism
  - Definition: Ring of Integers
  - Definition: Semigroup
  - Definition: Signum Function in An Ordered Field
  - Definition: Solution to a Lower Triangular SLE - Forward Substitution
  - Definition: Solution to an Upper Triangular SLE - Backward Substitution
  - Definition: Spectrum of a Commutative Ring
  - Definition: Square Matrix
  - Definition: Subadditive Function
  - Definition: Subfield
  - Definition: Subgroup
  - Definition: Subring
  - Definition: Subspace
  - Definition: Substructure
  - Definition: Symmetric Bilinear Form
  - Definition: Symmetric Matrix
  - Definition: Systems of Linear Equations with many Unknowns
  - Definition: Transcendental Element
  - Definition: Transposed Matrix
  - Definition: Unit
  - Definition: Unitary Affine Space
  - Definition: Upper and Lower Triangular Matrix
  - Definition: Vector Space
  - Definition: Zariski Topology of a Commutative Ring
  - Definition: Zero Divisor and Integral Domain
  - Definition: Zero Matrix, Zero Vector
  - Definition: Zero Ring
- Analysis (115)
  - Definition: (Weighted) Arithmetic Mean
  - Definition: Absolutely Convergent Complex Series
  - Definition: Absolutely Convergent Series
  - Definition: Accumulation Point (Real Numbers)
  - Definition: Accumulation Points (Complex Numbers)
  - Definition: Asymptotical Approximation
  - Definition: Banach Space
  - Definition: Bounded Complex Sequences
  - Definition: Bounded Complex Sets
  - Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
  - Definition: Bounded and Unbounded Functions
  - Definition: Closed Curve, Open Curve
  - Definition: Closed and Open Regions of the Complex Plane
  - Definition: Complete Metric Space
  - Definition: Complex Cauchy Sequence
  - Definition: Complex Infinite Series
  - Definition: Complex Polynomials
  - Definition: Complex Sequence
  - Definition: Constant Function Real Case
  - Definition: Continuous Complex Functions
  - Definition: Continuous Functions at Single Complex Numbers
  - Definition: Continuous Functions at Single Real Numbers
  - Definition: Continuous Real Functions
  - Definition: Continuously Differentiable Functions
  - Definition: Convergent Complex Sequence
  - Definition: Convergent Complex Series
  - Definition: Convergent Real Sequence
  - Definition: Convergent Real Series
  - Definition: Convex and Concave Functions
  - Definition: Cosine of a Real Variable
  - Definition: Curves In the Multidimensional Space `\(\mathbb R^n\)`
  - Definition: Decimal Representation of Real Numbers
  - Definition: Derivative of an n-Dimensional Curve
  - Definition: Derivative, Differentiable Functions
  - Definition: Difference Quotient
  - Definition: Directional Derivative
  - Definition: Divergent Sequences
  - Definition: Divergent Series
  - Definition: Even Complex Sequence
  - Definition: Even and Odd Complex Functions
  - Definition: Even and Odd Functions
  - Definition: Exponential Function of General Base
  - Definition: Extended Real Numbers
  - Definition: Finite and Sigma-Finite Measure
  - Definition: Finite and Sigma-Finite Pre-measure
  - Definition: First-Order Ordinary Differential Equation (ODE)
  - Definition: Functional
  - Definition: Functional Equation
  - Definition: Generalized Polynomial Function
  - Definition: Geometric Mean
  - Definition: Heine-Borel Property Defines Compact Subsets
  - Definition: Higher Order Directional Derivative
  - Definition: Higher-Order Derivatives
  - Definition: Hilbert Space
  - Definition: Hyperbolic Cosine
  - Definition: Hyperbolic Sine
  - Definition: Improper Integral
  - Definition: Infimum of Extended Real Numbers
  - Definition: Infimum, Greatest Lower Bound
  - Definition: Infinite Series, Partial Sums
  - Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
  - Definition: Isolated Point (Real Numbers)
  - Definition: Jordan Arc (Simple Curve)
  - Definition: Limit Inferior
  - Definition: Limit Superior
  - Definition: Limits of Complex Functions
  - Definition: Limits of Real Functions
  - Definition: Linear Function
  - Definition: Local Extremum
  - Definition: Logarithmically Convex and Concave Functions
  - Definition: Maximum (Real Numbers)
  - Definition: Measurable Set
  - Definition: Measurable Space
  - Definition: Measure
  - Definition: Measureable Function
  - Definition: Minimum (Real Numbers)
  - Definition: Monotonic Functions
  - Definition: Monotonic Sequences
  - Definition: Nested Real Intervals
  - Definition: Odd Complex Sequence
  - Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability
  - Definition: Open and Closed Discs
  - Definition: Order Relation for Step Functions
  - Definition: Periodic Functions
  - Definition: Pointwise and Uniformly Convergent Sequences of Functions
  - Definition: Polynomials
  - Definition: Positive and Negative Parts of a Real-Valued Function
  - Definition: Pre-measure
  - Definition: Rational Functions
  - Definition: Real Absolute Value Function
  - Definition: Real Cauchy Sequence
  - Definition: Real Identity Function
  - Definition: Real Intervals
  - Definition: Real Sequence
  - Definition: Real Subsequence
  - Definition: Rearrangement of Infinite Series
  - Definition: Reciprocal Function
  - Definition: Riemann Sum With Respect to a Partition
  - Definition: Riemann-Integrable Functions
  - Definition: Ring of Sets (measure-theoretic definition)
  - Definition: Sequences Tending To Infinity
  - Definition: Sigma-Algebra
  - Definition: Sine of a Real Variable
  - Definition: Solution of Ordinary DE
  - Definition: Step Functions
  - Definition: Supremum Norm for Functions
  - Definition: Supremum of Extended Real Numbers
  - Definition: Supremum, Least Upper Bound
  - Definition: Tangent of a Real Variable
  - Definition: Totally Differentiable Functions, Total Derivative
  - Definition: Uniformly Continuous Functions (Real Case)
  - Definition: Vector Field
  - Definition: `\(b\)`-Adic Fractions
  - Definition: `\(n\)` times Continuously Differentiable Functions
  - Definition: n-Periodical Complex Sequence
- Combinatorics (10)
  - Definition: Binomial Coefficients
  - Definition: Combinations
  - Definition: Cycles
  - Definition: Difference Operator
  - Definition: Factorial Polynomials
  - Definition: Falling And Rising Factorial Powers
  - Definition: Falling and Rising Factorial Powers of Functions
  - Definition: Indefinite Sum, Antidifference
  - Definition: Permutations
  - Definition: Stirling Numbers of First and Second Kind
- Geometry (160)
  - Analytic Geometry (4)
    - Definition: Describing a Straight Line Using Two Vectors
    - Definition: Hyperplane of a Number Space
    - Definition: Points in a Coordinate System - Number Spaces
    - Definition: Points vs. Vectors in a Number Space
  - Euclidean Geometry (150)
    - Definition: "Lies on" Relation
    - Definition: Congruence
    - Definition: Ellipse
    - Definition: Euclidean Movement - Isometry
    - Definition: Points, Straight Lines, and Planes
    - Definition: Similarity
    - Elements Euclid (144)
      - Book 1 Plane Geometry (35)
        
        Definition: 1.01: Point
        
        Definition: 1.02: Line, Curve
        
        Definition: 1.03: Intersections of Lines
        
        Definition: 1.04: Straight Line, Segment and Ray
        
        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
        
        Definition: 1.10: Right Angle, Perpendicular Straight Lines
        
        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
        
        Definition: 1.17: Diameter of the Circle
        
        Definition: 1.18: Semicircle
        
        Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure
        
        Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle
        
        Definition: 1.21: Right Triangle, Obtuse Triangle, Acute Triangle
        
        Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium
        
        Definition: 1.23: Parallel Straight Lines
        
        Definition: Altitude of a Triangle
        
        Definition: Collinear Points, Segments, Rays
        
        Definition: Concentric Circles
        
        Definition: Decagon
        
        Definition: Diagonal
        
        Definition: Exterior, Interior, Alternate and Corresponding Angles
        
        Definition: Hexagon
        
        Definition: Parallelogram - Defining Property III
        
        Definition: Pentagon
        
        Definition: Sum of Angles
        
        Definition: Supplemental Angles
        
        Definition: Triangle
      - Book 2 Geometric Algebra (3)
        
        Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides
        
        Definition: 2.2: Gnomon
        
        Definition: Point of Division, Point of External Division
      - Book 3 Circles (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
      - Book 4 Inscription And Circumscription (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: 4.5: Inscribing Circles in Rectilinear Figures
        
        Definition: 4.6: Circumscribing Circles about Rectilinear Figures
        
        Definition: 4.7: Chord and Secant
      - Book 5 Proportion (19)
        
        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
        
        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
        
        Definition: Geometric Progression, Continued Proportion
      - Book 6 Similar Figures (3)
        
        Definition: 6.01: Similar Rectilinear Figures
        
        Definition: 6.02: Cut in Extreme and Mean Ratio
        
        Definition: 6.03: Height of a Figure
      - Book 7 Elementary Number Theory (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
      - Book 10 Incommensurable Magnitudes (16)
        
        Definition: 10.02: Magnitudes Commensurable and Incommensurable in Square
        
        Definition: Def. 10.01: Magnitudes Commensurable and Incommensurable in Length
        
        Definition: Def. 10.03: Rational and Irrational Magnitudes
        
        Definition: Def. 10.04: Rational and Irrational Magnitudes in Square
        
        Definition: Def. 10.05: First Binomial
        
        Definition: Def. 10.06: Second Binomial
        
        Definition: Def. 10.07: Third Binomial
        
        Definition: Def. 10.08: Fourth Binomial
        
        Definition: Def. 10.09: Fifth Binomial
        
        Definition: Def. 10.10: Sixth Binomial
        
        Definition: Def. 10.11: First Apotome
        
        Definition: Def. 10.12: Second Apotome
        
        Definition: Def. 10.13: Third Apotome
        
        Definition: Def. 10.14: Fourth Apotome
        
        Definition: Def. 10.15: Fifth Apotome
        
        Definition: Def. 10.16: Sixth Apotome
      - Book 11 Elementary Stereometry (28)
        
        Definition: Def. 11.01: Solid Figures, Three-Dimensional Polyhedra
        
        Definition: Def. 11.02: Surface of a Solid Figure
        
        Definition: Def. 11.03: Straight Line at Right Angles To a Plane
        
        Definition: Def. 11.04: Plane at Right Angles to a Plane
        
        Definition: Def. 11.05: Inclination of a Straight Line to a Plane
        
        Definition: Def. 11.06: Inclination of a Plane to a Plane
        
        Definition: Def. 11.07: Similarly Inclined Planes
        
        Definition: Def. 11.08: Parallel Planes
        
        Definition: Def. 11.09: Similar Solid Figures
        
        Definition: Def. 11.10: Equal Solid Figures
        
        Definition: Def. 11.11: Solid Angle
        
        Definition: Def. 11.12: Pyramid, Tetrahedron
        
        Definition: Def. 11.13: Prism, Parallelepiped
        
        Definition: Def. 11.14: Sphere
        
        Definition: Def. 11.15: Axis of a Sphere
        
        Definition: Def. 11.16: Center of a Sphere
        
        Definition: Def. 11.17: Diameter of a Sphere
        
        Definition: Def. 11.18: Cone
        
        Definition: Def. 11.19: Axis of a Cone
        
        Definition: Def. 11.20: Base of a Cone
        
        Definition: Def. 11.21: Cylinder
        
        Definition: Def. 11.22: Axis of a Cylinder
        
        Definition: Def. 11.23: Bases of a Cylinder
        
        Definition: Def. 11.24: Similar Cones, Similar Cylinders
        
        Definition: Def. 11.25: Cube
        
        Definition: Def. 11.26: Octahedron
        
        Definition: Def. 11.27: Icosahedron
        
        Definition: Def. 11.28: Dodecahedron
  - Projective Geometry (6)
    - Definition: Collinear Points
    - Definition: Concurrent Straight Lines
    - Definition: Coplanar Points and Straight Lines
    - Definition: Incidence
    - Definition: Perspectivities
    - Definition: Projectivities, Ranges and Pencils
- Graph Theory (57)
  - Definition: Adjacency List Representation
  - Definition: Adjacency Matrix
  - Definition: Biconnected Graphs, `\(k\)`-Connected Graphs
  - Definition: Bipartite Graph
  - Definition: Chromatic Number and `$k$`-Coloring of a Graph
  - Definition: Closed Walks, Closed Trails, and Cycles
  - Definition: Complement Graph
  - Definition: Complete Bipartite Graph
  - Definition: Complete Graph
  - Definition: Connected Vertices
  - Definition: Connected and Disconnected Graphs, Bridges and Cutvertices
  - Definition: Cycle Graph
  - Definition: Cyclic, Acyclic Graph
  - Definition: Degree Sequence
  - Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph
  - Definition: Dual Planar Graph
  - Definition: Eulerian Graph
  - Definition: Eulerian Tour
  - Definition: Face Degree
  - Definition: Face, Infinite Face
  - Definition: Finite and Infinite Graphs
  - Definition: Girth and Circumference
  - Definition: Graph Decomposable Into `\(k\)` Trees
  - Definition: Hamiltonian Cycle
  - Definition: Hamiltonian Graph
  - Definition: Incidence, Adjacency, Neighbours
  - Definition: Incidence, Adjacency, Predecessor and Successor Vertices, Neighbours
  - Definition: Interlacing Pieces with Respect to a Cycle, Interlacement Graph
  - Definition: Isomorphic Digraphs
  - Definition: Isomorphic Undirected Graphs
  - Definition: Leaf
  - Definition: Minimal Tree Decomposability
  - Definition: Null Graph
  - Definition: Order of a Graph
  - Definition: Pieces of a Graph With Respect to A Cycle
  - Definition: Planar Drawing (Embedding)
  - Definition: Planar Graph
  - Definition: Regular Graph
  - Definition: Root, Degree of a Tree, Subtree, Height
  - Definition: Semi-Eulerian Graph
  - Definition: Semi-Eulerian Tour, Open Trail
  - Definition: Semi-Hamiltonian Graph
  - Definition: Semi-Hamiltonian Path
  - Definition: Separating and Non-Separating Cycles
  - Definition: Size of a Graph
  - Definition: Spanning Subgraph
  - Definition: Spanning Tree
  - Definition: Subdigraphs and Superdigraphs; Induced Subdigraph
  - Definition: Subdivision of a Graph
  - Definition: Subgraphs and Supergraphs; Induced Subgraph
  - Definition: Suppressing Vertices, Suppressed Multigraph
  - Definition: Trees and Forests
  - Definition: Undirected Graph, Vertices, Edges, Simple Graph
  - Definition: Vertex Degrees for Digraphs
  - Definition: Vertex Degrees for Undirected Graphs
  - Definition: Walks, Trails, and Paths
  - Definition: Weakly and Strongly Connected Digraphs
- Knot Theory (4)
  - Definition: Knot
  - Definition: Knot Diagram, Classical Crossing, Virtual Crossing
  - Definition: Reidemeister Moves, Planar Isotopy Moves, Diagrammatic Moves
  - Definition: Unknot
- Logic (45)
  - Definition: Atomic Formulae in Predicate Logic
  - Definition: Axioms
  - Definition: Boolean Algebra
  - Definition: Canonical Normal Form
  - Definition: Concatenation of Languages
  - Definition: Conjunction
  - Definition: Conjunctive and Disjunctive Canonical Normal Forms
  - Definition: Consistency and Negation-Completeness of a Logical Calculus
  - Definition: Contrapositive
  - Definition: Derivability Property
  - Definition: Disjunction
  - Definition: Domain of Discourse
  - Definition: Equivalence
  - Definition: Exclusive Disjunction
  - Definition: Formal Languages Generated From a Grammar
  - Definition: Function, Arity and Constant
  - Definition: Grammar (Syntax)
  - Definition: Implication
  - Definition: Interpretation of Propositions - the Law of the Excluded Middle
  - Definition: Interpretation of Strings of a Formal Language and Their Truth Function
  - Definition: Iteration of Languages, Kleene Star, Kleene Plus
  - Definition: Language
  - Definition: Literals, Minterms, and Maxterms
  - Definition: Logical Arguments
  - Definition: Logical Calculus
  - Definition: Negation
  - Definition: Negation of a String
  - Definition: Paradox
  - Definition: Predicate of a Logical Calculus
  - Definition: Proofs and Theorems in a Logical Calculus
  - Definition: Quantifier, Bound Variables, Free Variables
  - Definition: Rules of Inference
  - Definition: Satisfaction Relation, Model, Tautology, Contradiction
  - Definition: Semantics of PL0
  - Definition: Semantics of a Formal Language
  - Definition: Set of Truth Values (True and False)
  - Definition: Signature
  - Definition: Signature of Propositional Logic - PL0
  - Definition: Soundness and Completeness of a Logical Calculus
  - Definition: Strings (words) over an Alphabet
  - Definition: Syntax of PL0 - Propositions as Boolean Terms
  - Definition: Terms in Predicate Logic
  - Definition: Truth Table
  - Definition: Variable in a Logical Calculus
  - Definition: `$k$`-nary Connectives, Prime and Compound Propositions
- Number Systems Arithmetics (33)
  - Definition: Absolute Value of Complex Numbers
  - Definition: Absolute Value of Integers
  - Definition: Absolute Value of Rational Numbers
  - Definition: Absolute Value of Real Numbers (Modulus)
  - Definition: Addition of Complex Numbers
  - Definition: Argument of a Complex Number
  - Definition: Complex Conjugate
  - Definition: Convergent Rational Sequence
  - Definition: Definition of Complex Numbers
  - Definition: Definition of Irrational Numbers
  - Definition: Division of Real Numbers
  - Definition: Dot Product of Complex Numbers
  - Definition: Euler's Constant
  - Definition: Multiplication of Complex Numbers
  - Definition: Multiplication of Natural Numbers
  - Definition: Number `$\pi$`
  - Definition: Order Relation for Integers - Positive and Negative Integers
  - Definition: Order Relation for Natural Numbers
  - Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers
  - Definition: Order Relation of Real Numbers
  - Definition: Products
  - Definition: Ratio of Two Real Numbers
  - Definition: Rational Cauchy Sequence
  - Definition: Rational Sequence
  - Definition: Set of Natural Numbers (Peano)
  - Definition: Set-theoretic Definition of Order Relation for Natural Numbers
  - Definition: Set-theoretic Definitions of Natural Numbers
  - Definition: Subtraction of Complex Numbers
  - Definition: Subtraction of Integers
  - Definition: Subtraction of Rational Numbers
  - Definition: Subtraction of Real Numbers
  - Definition: Sums
  - Definition: Triangle Numbers
- Number Theory (33)
  - Definition: Arithmetic Function
  - Definition: Canonical Representation of Natural Numbers, Factorization
  - Definition: Canonical Representation of Positive Rational Numbers
  - Definition: Co-prime Numbers
  - Definition: Complete Residue System
  - Definition: Composite Number
  - Definition: Congruent, Residue
  - Definition: Diophantine Equations
  - Definition: Divisor, Complementary Divisor, Multiple
  - Definition: Divisor-Closed Sets
  - Definition: Euler function
  - Definition: Even and Odd Numbers
  - Definition: Floor and Ceiling Functions
  - Definition: Harmonic Series
  - Definition: Jacobi Symbol
  - Definition: Legendre Symbol
  - Definition: Modulo Operation for Real Numbers
  - Definition: Multiplicative Functions
  - Definition: Möbius Function, Square-free
  - Definition: Number of Divisors
  - Definition: Perfect Number
  - Definition: Perfect Square
  - Definition: Prime Numbers
  - Definition: Prime-Counting Function
  - Definition: Quadratic Residue, Quadratic Nonresidue
  - Definition: Reduced Residue System
  - Definition: Sets of Integers Co-Prime To a Given Integer
  - Definition: Subsets of Prime Numbers Not Dividing a Natural Number
  - Definition: Sum of Divisors
  - Definition: Twin Prime Numbers
  - Definition: Von Mangoldt Function
  - Sieve Methods (2)
    - Definition: Sieve of Eratosthenes
    - Definition: Sieve, Sieve Problem
- Probability Theory And Statistics (17)
  - Definition: Bernoulli Experiment
  - Definition: Certain and Impossible Event
  - Definition: Conditional Probability
  - Definition: Continuous Random Variables
  - Definition: Discrete Random Variables
  - Definition: Geometric Probability
  - Definition: Independent Events
  - Definition: Laplace Experiments and Elementary Events
  - Definition: Mutually Exclusive and Collectively Exhaustive Events
  - Definition: Mutually Independent Events
  - Definition: Pairwise Independent Events
  - Definition: Probability Distribution
  - Definition: Probability Mass Function
  - Definition: Probability and its Axioms
  - Definition: Random Experiments and Random Events
  - Definition: Random Variable, Realization, Population and Sample
  - Definition: Relative and Absolute Frequency
- Set Theory (63)
  - Definition: Bijective Function
  - Definition: Bounded Subsets of Ordered Sets
  - Definition: Bounded Subsets of Unordered Sets
  - Definition: Canonical Projection
  - Definition: Cartesian Product
  - Definition: Comparing the Elements of Posets and Chains
  - Definition: Comparison of Cardinal Numbers
  - Definition: Complete System of Representatives
  - Definition: Composition of Binary Relations
  - Definition: Constant Function
  - Definition: Contained Relation "`$\in_X$`"
  - Definition: Countable Set, Uncountable Set
  - Definition: Disjoint Sets
  - Definition: Embedding, Inclusion Map
  - Definition: Equipotent Sets
  - Definition: Equivalence Class
  - Definition: Equivalence Relation
  - Definition: Extensional Relation
  - Definition: Finite Set, Infinite Set
  - Definition: Fixed Point, Fixed Point Property
  - Definition: Generalized Union of Sets
  - Definition: Graph of a Function
  - Definition: Identity Function
  - Definition: Index Set and Set Family
  - Definition: Indicator (Characteristic) Function, Carrier
  - Definition: Inductive Set
  - Definition: Injective Function
  - Definition: Inverse Relation
  - Definition: Invertible Functions, Inverse Functions
  - Definition: Irreflexive, Asymmetric and Antisymmetric Binary Relations
  - Definition: Limit Ordinal
  - Definition: Minimal Inductive Set
  - Definition: Mostowski Function and Collapse
  - Definition: Mutually Disjoint Sets
  - Definition: Order Embedding
  - Definition: Ordered Pair, n-Tuple
  - Definition: Ordinal Number
  - Definition: Partial and Total Maps (Functions)
  - Definition: Power Set
  - Definition: Preorder, Partial Order and Poset
  - Definition: Quotient Set, Partition
  - Definition: Reflexive, Symmetric and Transitive Binary Relations
  - Definition: Relation
  - Definition: Restriction
  - Definition: Set Complement
  - Definition: Set Difference
  - Definition: Set Intersection
  - Definition: Set Partition
  - Definition: Set Union
  - Definition: Set, Set Element, Empty Set
  - Definition: Singleton
  - Definition: Special Elements of Ordered Sets
  - Definition: Strict Total Order, Strictly-ordered Set
  - Definition: Subset and Superset
  - Definition: Surjective Function
  - Definition: The Class of all Ordinals `$\Omega$`
  - Definition: Total Order and Chain
  - Definition: Total and Unique Binary Relations
  - Definition: Transitive Set
  - Definition: Universal Set
  - Definition: Well-founded Relation
  - Definition: Well-order, Well-ordered Set
  - Definition: Zero of a Function
- Theoretical Computer Science (22)
  - Complexity Theory (1)
    - Definition: Big O Notation
  - Computability (10)
    - Definition: Algorithm (Effective Procedure)
    - Definition: Computational Problem, Solution
    - Definition: GOTO Command, GOTO Program, Index
    - Definition: GOTO-Computable Functions
    - Definition: LOOP Command, LOOP Program
    - Definition: LOOP-Computable Functions
    - Definition: Unit-Cost Random Access Machine
    - Definition: WHILE Command, WHILE Program
    - Definition: WHILE-Computable Functions
    - Definition: `$\mathcal P$`-Computable and `$\mathcal P$`-Decidable Problems
  - Data Structures (1)
    - Definition: Linked List, List Nodes
  - Formal Languages (10)
    - Definition: Abstract Syntax Tree
    - Definition: Ambiguous and Unambiguous Grammars
    - Definition: Deterministic Finite Automaton (DFA)
    - Definition: Epsilon Non-Deteriministic Finite Automaton (`$\epsilon$`-NFA)
    - Definition: Equivalent Grammars
    - Definition: Non-deterministic Finite Automaton (NFA)
    - Definition: Type-0 (Phrase Structure) Grammars and Recursively Enumerable Languages
    - Definition: Type-1 (cs) Grammars and Context-sensitive Languages
    - Definition: Type-2 (cf) Grammars and Context-free Languages
    - Definition: Type-3 (Linear) Grammars and Regular Languages
- Theoretical Physics (7)
  - Classical Physics (2)
    - Definition: Average Velocity
    - Definition: Instantaneous Velocity
  - Special Relativity (5)
    - Definition: Frame of Reference
    - Definition: Inertial and Noninertial Frames of Reference
    - Definition: Meter
    - Definition: Second
    - Definition: Spacetime Diagram
- Topology (55)
  - Definition: Boundary Points, Closures, Interiors, and Exteriors
  - Definition: Bounded Sequence
  - Definition: Bounded Subset of a Metric Space
  - Definition: Carrier Set
  - Definition: Cauchy Sequence
  - Definition: Comparison of Filters, Finer and Coarser Filters
  - Definition: Continuous Function
  - Definition: Continuous Functions in Metric Spaces
  - Definition: Convergent Sequences and Limits
  - Definition: Cotangent Bundle
  - Definition: Dense Sets, Nowhere Dense Sets
  - Definition: Derived, Dense-in-itself, and Perfect Sets
  - Definition: Diameter In Metric Spaces
  - Definition: Differentiable Manifold, Atlas
  - Definition: Differential Form of Degree k
  - Definition: Discrete and Indiscrete Topology
  - Definition: First and Second Category Sets
  - Definition: Hereditary and Weakly Hereditary Properties
  - Definition: Homeomorphism, Homeomorphic Spaces
  - Definition: Isolated, Adherent, Limit, `$\omega$`-Accumulation and Condensation Points
  - Definition: Isometry
  - Definition: Limit of a Function
  - Definition: Limits and Accumulation Points of Sequences
  - Definition: Manifold
  - Definition: Metric (Distance)
  - Definition: Metric Space
  - Definition: Modulus of Continuity of a Continuous Function
  - Definition: Neighborhood
  - Definition: Norm, Normed Vector Space
  - Definition: Open Ball, Neighborhood
  - Definition: Open Cover
  - Definition: Open Function, Closed Function
  - Definition: Open Sets in Metric Spaces
  - Definition: Open and Closed Functions
  - Definition: Open, Closed, Clopen
  - Definition: Ordering of Topologies
  - Definition: Pointwise and Uniform Convergence
  - Definition: Regular Open, Regular Closed
  - Definition: Section over a Base Space
  - Definition: Sequence
  - Definition: Simplex
  - Definition: Subbasis and Basis of Topology
  - Definition: Subsequence
  - Definition: Tangent Bundle
  - Definition: Topological Chart
  - Definition: Topological Product, Product Topology
  - Definition: Topological Space, Topology
  - Definition: Topological Subspaces and Subspace Topologies
  - Definition: Topological Sum, Disjoint Union
  - Definition: Topological, Continuous, Open, and Closed Invariants
  - Definition: Transition Map
  - Definition: Ultrafilter
  - Definition: Uniformly Continuous Functions (General Metric Spaces Case)
  - Definition: `\(C^n\)` Differentiable Function
  - Definition: `\(C^{n}\)`-Diffeomorphism
Theorems (90)
- Algebra (12)
  - Theorem: Classification of Cyclic Groups
  - Theorem: Classification of Finite Groups with the Order of a Prime Number
  - Theorem: Connection between Rings, Ideals, and Fields
  - Theorem: Construction of Fields from Integral Domains
  - Theorem: Construction of Groups from Commutative and Cancellative Semigroups
  - Theorem: Finite Basis Theorem
  - Theorem: Finite Integral Domains are Fields
  - Theorem: First Isomorphism Theorem for Groups
  - Theorem: Isomorphism of Rings
  - Theorem: Order of Cyclic Group (Fermat's Little Theorem)
  - Theorem: Order of Subgroup Divides Order of Finite Group
  - Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs
- Analysis (28)
  - Theorem: Bernoulli's Inequality
  - Theorem: Completeness Principle for Complex Numbers
  - Theorem: Completeness Principle for Real Numbers
  - Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains
  - Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous
  - Theorem: Darboux's Theorem
  - Theorem: De Moivre's Identity, Complex Powers
  - Theorem: Defining Properties of the Field of Real Numbers
  - Theorem: Every Bounded Monotonic Sequence Is Convergent
  - Theorem: Every Bounded Real Sequence has a Convergent Subsequence
  - Theorem: Fundamental Theorem of Calculus
  - Theorem: Heine-Borel Theorem
  - Theorem: Indefinite Integral, Antiderivative
  - Theorem: Inequality Between the Geometric and the Arithmetic Mean
  - Theorem: Inequality of Weighted Arithmetic Mean
  - Theorem: Inequality of the Arithmetic Mean
  - Theorem: Integration by Substitution
  - Theorem: Intermediate Root Value Theorem
  - Theorem: Intermediate Value Theorem
  - Theorem: Mean Value Theorem For Riemann Integrals
  - Theorem: Nested Closed Subset Theorem
  - Theorem: Partial Integration
  - Theorem: Reverse Triangle Inequalities
  - Theorem: Rolle's Theorem
  - Theorem: Squeezing Theorem for Functions
  - Theorem: Supremum Property, Infimum Property
  - Theorem: Taylor's Formula
  - Theorem: Triangle Inequality
- Combinatorics (6)
  - Theorem: Approximation of Factorials Using the Stirling Formula
  - Theorem: Binomial Theorem
  - Theorem: Fundamental Theorem of the Difference Calculus
  - Theorem: Inclusion-Exclusion Principle (Sylvester's Formula)
  - Theorem: Multinomial Theorem
  - Theorem: Taylor's Formula Using the Difference Operator
- Geometry (2)
  - Analytic Geometry (1)
    - Theorem: Existence and Uniqueness of a Straight Line Through Two Points
  - Euclidean Geometry (1)
    - Elements Euclid (1)
      - Book 9 Number Theory Applications (1)
        
        Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic
- Graph Theory (13)
  - Theorem: Brooks' Theorem
  - Theorem: Characterization of Biconnected Planar Graphs
  - Theorem: Characterization of Bipartite Graphs
  - Theorem: Characterization of Eulerian Graphs
  - Theorem: Characterization of Planar Graphs
  - Theorem: Characterization of Planar Hamiltonian Graphs
  - Theorem: Characterization of Semi-Eulerian Graphs
  - Theorem: Euler Characteristic for Planar Graphs
  - Theorem: Five Color Theorem for Planar Graphs
  - Theorem: Four Color Theorem for Planar Graphs
  - Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph
  - Theorem: Number of Labeled Spanning Trees
  - Theorem: Six Color Theorem for Planar Graphs
- Number Theory (12)
  - Theorem: Chinese Remainder Theorem
  - Theorem: Commutative Group of Multiplicative Functions
  - Theorem: Euler-Fermat Theorem
  - Theorem: Fermat's Last Theorem
  - Theorem: First Supplementary Law to the Quadratic Reciprocity Law
  - Theorem: Fundamental Theorem of Arithmetic
  - Theorem: Infinite Set of Prime Numbers
  - Theorem: Möbius Inversion Formula
  - Theorem: Number of Multiples of a Prime Number Less Than Factorial
  - Theorem: Properties of the Jacobi Symbol
  - Theorem: Quadratic Reciprocity Law
  - Theorem: Second Supplementary Law to the Quadratic Reciprocity Law
- Probability Theory And Statistics (3)
  - Theorem: Bayes' Theorem
  - Theorem: Law of Total Probability
  - Theorem: Theorem of Large Numbers for Relative Frequencies
- Set Theory (4)
  - Theorem: Distinction Between Finite and Infinite Sets Using Subsets
  - Theorem: Mostowski's Theorem
  - Theorem: Schröder-Bernstein Theorem
  - Theorem: Trichotomy of Ordinals
- Theoretical Computer Science (5)
  - Computability (2)
    - Theorem: Simulating LOOP Programs Using WHILE Programs
    - Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa)
  - Formal Languages (3)
    - Theorem: Deterministic Finite Automata Describe Regular Languages
    - Theorem: Reduction of NFA to DFA (Rabin-Scott Theorem)
    - Theorem: Reduction of `$\epsilon$`-NFA to NFA
- Theoretical Physics (3)
  - Classical Physics (3)
    - Theorem: First Law of Planetary Motion
    - Theorem: Second Law of Planetary Motion
    - Theorem: Third Law of Planetary Motion
- Topology (2)
  - Theorem: Continuous Functions on Compact Domains are Uniformly Continuous
  - Theorem: Theorem of Bolzano-Weierstrass
Lemmas (124)
- Algebra (22)
  - Lemma: A Criterion for Associates
  - Lemma: Any Positive Characteristic Is a Prime Number
  - Lemma: Continuants and Convergents
  - Lemma: Cyclic Groups are Abelian
  - Lemma: Divisibility of Principal Ideals
  - Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE
  - Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace
  - Lemma: Factor Groups
  - Lemma: Factor Rings, Generalization of Congruence Classes
  - Lemma: Fiber of Maximal Ideals
  - Lemma: Fiber of Prime Ideals
  - Lemma: Fiber of Prime Ideals Under a Spectrum Function
  - Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations
  - Lemma: Greatest Common Divisor and Least Common Multiple of Ideals
  - Lemma: Group Homomorphisms and Normal Subgroups
  - Lemma: Kernel and Image of Group Homomorphism
  - Lemma: Kernel and Image of a Group Homomorphism are Subgroups
  - Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring
  - Lemma: Prime Ideals of Multiplicative Systems in Integral Domains
  - Lemma: Subgroups and Their Cosets are Equipotent
  - Lemma: Subgroups of Cyclic Groups
  - Lemma: Uniqueness Lemma of a Finite Basis
- Analysis (14)
  - Lemma: Abel's Lemma for Testing Convergence
  - Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals
  - Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions
  - Lemma: Convergence Test for Telescoping Series
  - Lemma: Decreasing Sequence of Suprema of Extended Real Numbers
  - Lemma: Euler's Identity
  - Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point
  - Lemma: Increasing Sequence of Infima of Extended Real Numbers
  - Lemma: Invertible Functions on Real Intervals
  - Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine
  - Lemma: Sum of Roots Of Unity in Complete Residue Systems
  - Lemma: Trapezoid Rule
  - Lemma: Unit Circle
  - Lemma: Upper Bound for the Product of General Powers
- Combinatorics (1)
  - Lemma: Stirling Numbers and Rising Factorial Powers
- Geometry (16)
  - Analytic Geometry (1)
    - Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors
  - Euclidean Geometry (15)
    - Elements Euclid (15)
      - Book 10 Incommensurable Magnitudes (9)
        
        Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
        
        Lemma: Lem. 10.021: Medial is Irrational
        
        Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square
        
        Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square
        
        Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II
        
        Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational
        
        Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
        
        Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them
        
        Lemma: Lem. 10.13: Finding Pythagorean Magnitudes
      - Book 11 Elementary Stereometry (1)
        
        Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares
      - Book 12 Proportional Stereometry (2)
        
        Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters
        
        Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
      - Book 13 Platonic Solids (3)
        
        Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
        
        Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere
        
        Lemma: Lem. 13.18: Angle of the Pentagon
- Graph Theory (11)
- Logic (26)
  - Lemma: A Criterion for Valid Logical Arguments
  - Lemma: A proposition cannot be both, true and false
  - Lemma: A proposition cannot be equivalent to its negation
  - Lemma: Affirming the Consequent of an Implication
  - Lemma: Boolean Algebra of Propositional Logic
  - Lemma: Boolean Function
  - Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms
  - Lemma: De Morgan's Laws (Logic)
  - Lemma: Denying the Antecedent of an Implication
  - Lemma: Disjunctive Syllogism
  - Lemma: Distributivity of Conjunction and Disjunction
  - Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition
  - Lemma: Every Proposition Implies Itself
  - Lemma: Hypothetical Syllogism
  - Lemma: Implication as a Disjunction
  - Lemma: It is true that something can be (either) true or false
  - Lemma: Mixing-up the Inclusive and Exclusive Disjunction
  - Lemma: Mixing-up the Sufficient and Necessary Conditions
  - Lemma: Modus Ponens
  - Lemma: Modus Tollens
  - Lemma: Negation of an Implication
  - Lemma: The Proving Principle By Contraposition, Contrapositive
  - Lemma: The Proving Principle by Complete Induction
  - Lemma: The Proving Principle by Contradiction
  - Lemma: The Proving Principle by Transfinite Induction
  - Lemma: Unique Valuation of Minterms and Maxterms
- Number Systems Arithmetics (8)
- Number Theory (11)
  - Lemma: Coprimality and Congruence Classes
  - Lemma: Division with Quotient and Remainder
  - Lemma: Gaussian Lemma (Number Theory)
  - Lemma: Generalized Euclidean Lemma
  - Lemma: Möbius and Floor Functions Combined
  - Lemma: Reciprocity Law for Floor Functions
  - Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed
  - Lemma: Sum of Möbius Function Over Divisors With Division
  - Lemma: Sums of Floors
  - Lemma: Upper Bound of Harmonic Series Times Möbius Function
  - Sieve Methods (1)
    - Lemma: Sieve for Twin Primes
- Set Theory (11)
  - Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull
  - Lemma: Behavior of Functions with Set Operations
  - Lemma: Comparing the Elements of Strictly Ordered Sets
  - Lemma: Composition of Functions
  - Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property
  - Lemma: Composition of Relations Preserves Their Right-Uniqueness Property
  - Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals
  - Lemma: Finite Cardinal Numbers and Set Operations
  - Lemma: Properties of Ordinal Numbers
  - Lemma: Successor of Ordinal
  - Lemma: Zorn's Lemma
- Theoretical Computer Science (1)
  - Computability (1)
    - Lemma: LOOP-Computable Functions are Total
- Topology (3)
Propositions (1063)
- Algebra (32)
  - Proposition: A Field with an Absolute Value is a Metric Space
  - Proposition: Abelian Group of Matrices Under Addition
  - Proposition: Additive Subgroups of Integers
  - Proposition: Cancellation Law
  - Proposition: Characterization of Dependent Absolute Values
  - Proposition: Characterization of Non-Archimedean Absolute Values
  - Proposition: Criteria for Subgroups
  - Proposition: Criterions for Equality of Principal Ideals
  - Proposition: Finite Order of an Element Equals Order Of Generated Group
  - Proposition: Generalization of Cancellative Multiplication of Integers
  - Proposition: Group Homomorphisms with Cyclic Groups
  - Proposition: Group of Units
  - Proposition: In a Field, `$0$` Is Unequal `$1$`
  - Proposition: Open and Closed Subsets of a Zariski Topology
  - Proposition: Principal Ideal Generated by A Unit
  - Proposition: Principal Ideals being Maximal Ideals
  - Proposition: Principal Ideals being Prime Ideals
  - Proposition: Properties of Cosets
  - Proposition: Properties of a Complex Scalar Product
  - Proposition: Properties of a Group Homomorphism
  - Proposition: Quadratic Formula
  - Proposition: Quotient Space
  - Proposition: Simple Calculations Rules in a Group
  - Proposition: Simple Consequences from the Definition of a Vector Space
  - Proposition: Spectrum Function of Commutative Rings
  - Proposition: Square of a Non-Zero Element is Positive in Ordered Fields
  - Proposition: Subgroups of Finite Cyclic Groups
  - Proposition: Subset of Powers is a Submonoid
  - Proposition: Unique Solvability of `$a\ast x=b$` in Groups
  - Proposition: Uniqueness of Inverse Elements
  - Proposition: Uniqueness of the Neutral Element
  - Proposition: `$0$` Is Less Than `$1$` In Ordered Fields
- Analysis (222)
  - Proposition: A General Criterion for the Convergence of Infinite Complex Series
  - Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions
  - Proposition: Abel's Test
  - Proposition: Additivity Theorem of Tangent
  - Proposition: Additivity Theorems of Cosine and Sine
  - Proposition: Antiderivatives are Uniquely Defined Up to a Constant
  - Proposition: Approximation of Functions by Taylor's Formula
  - Proposition: Arithmetic of Functions with Limits - Difference
  - Proposition: Arithmetic of Functions with Limits - Division
  - Proposition: Arithmetic of Functions with Limits - Product
  - Proposition: Arithmetic of Functions with Limits - Sums
  - Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule
  - Proposition: Bounds for Partial Sums of Exponential Series
  - Proposition: Calculation Rules for General Powers
  - Proposition: Calculations with Uniformly Convergent Functions
  - Proposition: Cauchy Condensation Criterion
  - Proposition: Cauchy Criterion
  - Proposition: Cauchy Product of Absolutely Convergent Complex Series
  - Proposition: Cauchy Product of Absolutely Convergent Series
  - Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent
  - Proposition: Cauchy-Schwarz Inequality for Integral p-norms
  - Proposition: Cauchy-Schwarz Test
  - Proposition: Cauchy–Schwarz Inequality
  - Proposition: Chain Rule
  - Proposition: Characterization of Monotonic Functions via Derivatives
  - Proposition: Closed Formula for the Maximum and Minimum of Two Numbers
  - Proposition: Closed Subsets of Compact Sets are Compact
  - Proposition: Closed n-Dimensional Cuboids Are Compact
  - Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum
  - Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed
  - Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth
  - Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences
  - Proposition: Complex Conjugate of Complex Exponential Function
  - Proposition: Complex Convergent Sequences are Bounded
  - Proposition: Complex Exponential Function
  - Proposition: Composition of Continuous Functions at a Single Point
  - Proposition: Compositions of Continuous Functions on a Whole Domain
  - Proposition: Continuity of Complex Exponential Function
  - Proposition: Continuity of Cosine and Sine
  - Proposition: Continuity of Exponential Function
  - Proposition: Continuity of Exponential Function of General Base
  - Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals
  - Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable
  - Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity
  - Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero
  - Proposition: Convergence Behavior of the Sequence `\((b^n)\)`
  - Proposition: Convergence Behaviour of Absolutely Convergent Series
  - Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero
  - Proposition: Convergent Complex Sequences Are Bounded
  - Proposition: Convergent Complex Sequences Are Cauchy Sequences
  - Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences
  - Proposition: Convergent Real Sequences Are Cauchy Sequences
  - Proposition: Convergent Real Sequences are Bounded
  - Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space
  - Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space
  - Proposition: Convergent Sequences are Bounded
  - Proposition: Convex Functions on Open Intervals are Continuous
  - Proposition: Convexity and Concaveness Test
  - Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm
  - Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`
  - Proposition: Derivative of Cosine
  - Proposition: Derivative of General Powers of Positive Numbers
  - Proposition: Derivative of Sine
  - Proposition: Derivative of Tangent
  - Proposition: Derivative of an Invertible Function on Real Invervals
  - Proposition: Derivative of the Exponential Function
  - Proposition: Derivative of the Inverse Sine
  - Proposition: Derivative of the Inverse Tangent
  - Proposition: Derivative of the Natural Logarithm
  - Proposition: Derivative of the Reciprocal Function
  - Proposition: Derivative of the n-th Power Function
  - Proposition: Derivatives of Even and Odd Functions
  - Proposition: Difference of Convergent Complex Sequences
  - Proposition: Difference of Convergent Real Sequences
  - Proposition: Difference of Convergent Real Series
  - Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine
  - Proposition: Differentiable Functions and Tangent-Linear Approximation
  - Proposition: Differential Equation of the Exponential Function
  - Proposition: Direct Comparison Test For Absolutely Convergent Complex Series
  - Proposition: Direct Comparison Test For Absolutely Convergent Series
  - Proposition: Direct Comparison Test For Divergent Series
  - Proposition: Dirichlet's Test
  - Proposition: Estimate for the Remainder Term of Complex Exponential Function
  - Proposition: Estimate for the Remainder Term of Exponential Function
  - Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine
  - Proposition: Euler's Formula
  - Proposition: Eveness (Oddness) of Polynomials
  - Proposition: Eveness of the Cosine of a Real Variable
  - Proposition: Exponential Function
  - Proposition: Exponential Function of General Base With Integer Exponents
  - Proposition: Exponential Function of General Base With Natural Exponents
  - Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals
  - Proposition: Functional Equation of the Complex Exponential Function
  - Proposition: Functional Equation of the Exponential Function
  - Proposition: Functional Equation of the Exponential Function of General Base
  - Proposition: Functional Equation of the Exponential Function of General Base (Revised)
  - Proposition: Functional Equation of the Natural Logarithm
  - Proposition: Gamma Function
  - Proposition: Gamma Function Interpolates the Factorial
  - Proposition: General Powers of Positive Numbers
  - Proposition: Generalized Bernoulli's Inequality
  - Proposition: Generalized Product Rule
  - Proposition: Generalized Triangle Inequality
  - Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members
  - Proposition: How Convergence Preserves the Order Relation of Sequence Members
  - Proposition: Hölder's Inequality
  - Proposition: Hölder's Inequality for Integral p-norms
  - Proposition: Identity Function is Continuous
  - Proposition: Image of a Compact Set Under a Continuous Function
  - Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials
  - Proposition: Inequality between Powers of `$2$` and Factorials
  - Proposition: Inequality between Square Numbers and Powers of `$2$`
  - Proposition: Infinite Geometric Series
  - Proposition: Infinite Series for Cosine and Sine
  - Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function
  - Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function
  - Proposition: Integral Test for Convergence
  - Proposition: Integral of Cosine
  - Proposition: Integral of General Powers
  - Proposition: Integral of Inverse Sine
  - Proposition: Integral of Sine
  - Proposition: Integral of the Exponential Function
  - Proposition: Integral of the Inverse Tangent
  - Proposition: Integral of the Natural Logarithm
  - Proposition: Integral of the Reciprocal Function
  - Proposition: Integrals on Adjacent Intervals
  - Proposition: Inverse Cosine of a Real Variable
  - Proposition: Inverse Hyperbolic Cosine
  - Proposition: Inverse Hyperbolic Sine
  - Proposition: Inverse Sine of a Real Variable
  - Proposition: Inverse Tangent and Complex Exponential Function
  - Proposition: Inverse Tangent of a Real Variable
  - Proposition: Legendre Polynomials and Legendre Differential Equations
  - Proposition: Leibniz Criterion for Alternating Series
  - Proposition: Limit Comparizon Test
  - Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence
  - Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence
  - Proposition: Limit Test for Roots or Ratios
  - Proposition: Limit of 1/n
  - Proposition: Limit of Exponential Growth as Compared to Polynomial Growth
  - Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth
  - Proposition: Limit of Nested Real Intervals
  - Proposition: Limit of Nth Powers
  - Proposition: Limit of Nth Root of N
  - Proposition: Limit of Nth Root of a Positive Constant
  - Proposition: Limit of a Function is Unique If It Exists
  - Proposition: Limit of a Polynomial
  - Proposition: Limit of a Rational Function
  - Proposition: Limit of the Constant Function
  - Proposition: Limit of the Identity Function
  - Proposition: Limits of General Powers
  - Proposition: Limits of Logarithm in `$[0,+\infty]$`
  - Proposition: Limits of Polynomials at Infinity
  - Proposition: Linearity and Monotony of the Riemann Integral
  - Proposition: Linearity and Monotony of the Riemann Integral for Step Functions
  - Proposition: Logarithm to a General Base
  - Proposition: Minkowski's Inequality
  - Proposition: Minkowski's Inequality for Integral p-norms
  - Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable
  - Proposition: Monotony Criterion
  - Proposition: Natural Logarithm
  - Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers
  - Proposition: Not all Continuous Functions are also Uniformly Continuous
  - Proposition: Nth Powers
  - Proposition: Nth Roots of Positive Numbers
  - Proposition: Oddness of the Sine of a Real Variable
  - Proposition: Only the Uniform Convergence Preserves Continuity
  - Proposition: Open Intervals Contain Uncountably Many Irrational Numbers
  - Proposition: Open Real Intervals are Uncountable
  - Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable
  - Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions
  - Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain
  - Proposition: Preservation of Inequalities for Limits of Functions
  - Proposition: Product of Convegent Complex Sequences
  - Proposition: Product of Convegent Real Sequences
  - Proposition: Product of Riemann-integrable Functions is Riemann-integrable
  - Proposition: Product of a Complex Number and a Convergent Complex Sequence
  - Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity
  - Proposition: Product of a Real Number and a Convergent Real Sequence
  - Proposition: Product of a Real Number and a Convergent Real Series
  - Proposition: Pythagorean Identity
  - Proposition: Quotient of Convergent Complex Sequences
  - Proposition: Quotient of Convergent Real Sequences
  - Proposition: Raabe's Test
  - Proposition: Ratio Test
  - Proposition: Ratio Test For Absolutely Convergent Complex Series
  - Proposition: Rational Functions are Continuous
  - Proposition: Rational Numbers are Dense in Real Numbers
  - Proposition: Rational Powers of Positive Numbers
  - Proposition: Real Sequences Contain Monotonic Subsequences
  - Proposition: Rearrangement of Absolutely Convergent Series
  - Proposition: Rearrangement of Convergent Series
  - Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence
  - Proposition: Riemann Integral for Step Functions
  - Proposition: Riemann Sum Converging To the Riemann Integral
  - Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions
  - Proposition: Root Test
  - Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function
  - Proposition: Square Roots
  - Proposition: Step Function on Closed Intervals are Riemann-Integrable
  - Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval
  - Proposition: Sufficient Condition for a Local Extremum
  - Proposition: Sum of Arguments of Hyperbolic Cosine
  - Proposition: Sum of Arguments of Hyperbolic Sine
  - Proposition: Sum of Convergent Complex Sequences
  - Proposition: Sum of Convergent Real Sequences
  - Proposition: Sum of Convergent Real Series
  - Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty
  - Proposition: Supremum Norm and Uniform Convergence
  - Proposition: Taylor's Formula with Remainder Term of Lagrange
  - Proposition: The distance of complex numbers makes complex numbers a metric space.
  - Proposition: The distance of real numbers makes real numbers a metric space.
  - Proposition: Uniform Convergence Criterion of Cauchy
  - Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series
  - Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions
  - Proposition: Uniqueness Of the Limit of a Sequence
  - Proposition: Zero of Cosine
  - Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum
  - Proposition: `\(\exp(0)=1\)`
  - Proposition: `\(\exp(0)=1\)` (Complex Case)
  - Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences
  - Proposition: n-th Roots of Unity
- Combinatorics (26)
  - Proposition: Antidifferences are Unique Up to a Periodic Constant
  - Proposition: Antidifferences of Some Functions
  - Proposition: Basic Calculations Involving Indefinite Sums
  - Proposition: Basic Calculations Involving the Difference Operator
  - Proposition: Closed Formula For Binomial Coefficients
  - Proposition: Comparison between the Stirling numbers of the First and Second Kind
  - Proposition: Difference Operator of Falling Factorial Powers
  - Proposition: Difference Operator of Powers
  - Proposition: Factorial
  - Proposition: Factorial Polynomials have a Unique Representation
  - Proposition: Factorial Polynomials vs. Polynomials
  - Proposition: Factorials and Stirling Numbers of the First Kind
  - Proposition: Fundamental Counting Principle
  - Proposition: Indicator Function and Set Operations
  - Proposition: Inversion Formulas For Stirling Numbers
  - Proposition: Multinomial Coefficient
  - Proposition: Nth Difference Operator
  - Proposition: Number of Ordered n-Tuples in a Set
  - Proposition: Number of Relations on a Finite Set
  - Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters
  - Proposition: Number of Subsets of a Finite Set
  - Proposition: Recursive Formula for Binomial Coefficients
  - Proposition: Recursive Formula for the Stirling Numbers of the First Kind
  - Proposition: Recursive Formula for the Stirling Numbers of the Second Kind
  - Proposition: Recursively Defined Arithmetic Functions, Recursion
  - Proposition: Simple Binomial Identities
- Geometry (469)
  - Analytic Geometry (1)
    - Proposition: Presentation of a Straight Line in a Plane as a Linear Equation
  - Euclidean Geometry (468)
    - Proposition: Common Points of Two Distinct Straight Lines in a Plane
    - Proposition: Common Points of a Plane and a Straight Line Not in the Plane
    - Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line
    - Proposition: Plane Determined by two Crossing Straight Lines
    - Elements Euclid (464)
      - Book 1 Plane Geometry (48)
        
        Proposition: 1.01: Constructing an Equilateral Triangle
        
        Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment
        
        Proposition: 1.03: Cutting a Segment at a Given Size
        
        Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
        
        Proposition: 1.05: Isosceles Triangles I
        
        Proposition: 1.06: Isosceles Triagles II
        
        Proposition: 1.07: Uniqueness of Triangles
        
        Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
        
        Proposition: 1.09: Bisecting an Angle
        
        Proposition: 1.10: Bisecting a Segment
        
        Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line
        
        Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
        
        Proposition: 1.13: Angles at Intersections of Straight Lines
        
        Proposition: 1.14: Combining Rays to Straight Lines
        
        Proposition: 1.15: Opposite Angles on Intersecting Straight Lines
        
        Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
        
        Proposition: 1.17: The Sum of Two Angles of a Triangle
        
        Proposition: 1.18: Angles and Sides in a Triangle I
        
        Proposition: 1.19: Angles and Sides in a Triangle II
        
        Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
        
        Proposition: 1.21: Triangles within Triangles
        
        Proposition: 1.22: Construction of Triangles From Arbitrary Segments
        
        Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
        
        Proposition: 1.24: Angles and Sides in a Triangle III
        
        Proposition: 1.25: Angles and Sides in a Triangle IV
        
        Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
        
        Proposition: 1.27: Parallel Lines I
        
        Proposition: 1.28: Parallel Lines II
        
        Proposition: 1.29: Parallel Lines III
        
        Proposition: 1.30: Transitivity of Parallel Lines
        
        Proposition: 1.31: Constructing a Parallel Line from a Line and a Point
        
        Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle
        
        Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
        
        Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
        
        Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels
        
        Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels
        
        Proposition: 1.37: Triangles of Equal Area I
        
        Proposition: 1.38: Triangles of Equal Area II
        
        Proposition: 1.39: Triangles of Equal Area III
        
        Proposition: 1.40: Triangles of Equal Area IV
        
        Proposition: 1.41: Parallelograms and Triagles
        
        Proposition: 1.42: Construction of Parallelograms I
        
        Proposition: 1.43: Complementary Segments of Parallelograms
        
        Proposition: 1.44: Construction of Parallelograms II
        
        Proposition: 1.45: Construction of Parallelograms III
        
        Proposition: 1.46: Construction of a Square on a Given Segment
        
        Proposition: 1.47: Pythagorean Theorem
        
        Proposition: 1.48: The Converse of the Pythagorean Theorem
      - Book 2 Geometric Algebra (14)
        
        Proposition: 2.01: Summing Areas or Rectangles
        
        Proposition: 2.02: Square is Sum of Two Rectangles
        
        Proposition: 2.03: Rectangle is Sum of Square and Rectangle
        
        Proposition: 2.04: Square of Sum
        
        Proposition: 2.05: Rectangle is Difference of Two Squares
        
        Proposition: 2.06: Square of Sum with One Halved Summand
        
        Proposition: 2.07: Sum of Squares
        
        Proposition: 2.08: Square of Sum with One Doubled Summand
        
        Proposition: 2.09: Sum of Squares of Sum and Difference
        
        Proposition: 2.10: Sum of Squares (Half)
        
        Proposition: 2.11: Constructing the Golden Ratio of a Segment
        
        Proposition: 2.12: Law of Cosines (for Obtuse Angles)
        
        Proposition: 2.13: Law of Cosines (for Acute Angles)
        
        Proposition: 2.14: Constructing a Square from a Rectilinear Figure
      - Book 3 Circles (37)
        
        Proposition: 3.01: Finding the Center of a given Circle
        
        Proposition: 3.02: Chord Lies Inside its Circle
        
        Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector
        
        Proposition: 3.04: Chords do not Bisect Each Other
        
        Proposition: 3.05: Intersecting Circles have Different Centers
        
        Proposition: 3.06: Touching Circles have Different Centers
        
        Proposition: 3.07: Relative Lengths of Lines Inside Circle
        
        Proposition: 3.08: Relative Lengths of Lines Outside Circle
        
        Proposition: 3.09: Condition for Point to be Center of Circle
        
        Proposition: 3.10: Two Circles have at most Two Points of Intersection
        
        Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally
        
        Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally
        
        Proposition: 3.13: Circles Touch at One Point at Most
        
        Proposition: 3.14: Equal Chords in Circle
        
        Proposition: 3.15: Relative Lengths of Chords of Circles
        
        Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle
        
        Proposition: 3.17: Construction of Tangent from Point to Circle
        
        Proposition: 3.18: Radius at Right Angle to Tangent
        
        Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center
        
        Proposition: 3.20: Inscribed Angle Theorem
        
        Proposition: 3.21: Angles in Same Segment of Circle are Equal
        
        Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral
        
        Proposition: 3.23: Segment on Given Base Unique
        
        Proposition: 3.24: Similar Segments on Equal Bases are Equal
        
        Proposition: 3.25: Construction of Circle from Segment
        
        Proposition: 3.26: Equal Angles and Arcs in Equal Circles
        
        Proposition: 3.27: Angles on Equal Arcs are Equal
        
        Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles
        
        Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines
        
        Proposition: 3.30: Bisection of Arc
        
        Proposition: 3.31: Relative Sizes of Angles in Segments
        
        Proposition: 3.32: Angles made by Chord with Tangent
        
        Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle
        
        Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle
        
        Proposition: 3.35: Intersecting Chord Theorem
        
        Proposition: 3.36: Tangent Secant Theorem
        
        Proposition: 3.37: Converse of Tangent Secant Theorem
      - Book 4 Inscription And Circumscription (16)
        
        Proposition: 4.01: Fitting Chord Into Circle
        
        Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles
        
        Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles
        
        Proposition: 4.04: Inscribing Circle in Triangle
        
        Proposition: 4.05: Circumscribing Circle about Triangle
        
        Proposition: 4.06: Inscribing Square in Circle
        
        Proposition: 4.07: Circumscribing Square about Circle
        
        Proposition: 4.08: Inscribing Circle in Square
        
        Proposition: 4.09: Circumscribing Circle about Square
        
        Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
        
        Proposition: 4.11: Inscribing Regular Pentagon in Circle
        
        Proposition: 4.12: Circumscribing Regular Pentagon about Circle
        
        Proposition: 4.13: Inscribing Circle in Regular Pentagon
        
        Proposition: 4.14: Circumscribing Circle about Regular Pentagon
        
        Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle
        
        Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle
      - Book 5 Proportion (25)
        
        Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition
        
        Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition
        
        Proposition: 5.03: Multiplication of Numbers is Associative
        
        Proposition: 5.04: Multiples of Terms in Equal Ratios
        
        Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction
        
        Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction
        
        Proposition: 5.07: Ratios of Equal Magnitudes
        
        Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes
        
        Proposition: 5.09: Magnitudes with Same Ratios are Equal
        
        Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios
        
        Proposition: 5.11: Equality of Ratios is Transitive
        
        Proposition: 5.12: Sum of Components of Equal Ratios
        
        Proposition: 5.13: Relative Sizes of Proportional Magnitudes
        
        Proposition: 5.14: Relative Sizes of Components of Ratios
        
        Proposition: 5.15: Ratio Equals its Multiples
        
        Proposition: 5.16: Proportional Magnitudes are Proportional Alternately
        
        Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated
        
        Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded
        
        Proposition: 5.19: Proportional Magnitudes have Proportional Remainders
        
        Proposition: 5.20: Relative Sizes of Successive Ratios
        
        Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion
        
        Proposition: 5.22: Equality of Ratios Ex Aequali
        
        Proposition: 5.23: Equality of Ratios in Perturbed Proportion
        
        Proposition: 5.24: Sum of Antecedents of Proportion
        
        Proposition: 5.25: Sum of Antecedent and Consequent of Proportion
      - Book 6 Similar Figures (33)
        
        Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base
        
        Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally
        
        Proposition: 6.03: Angle Bisector Theorem
        
        Proposition: 6.04: Equiangular Triangles are Similar
        
        Proposition: 6.05: Triangles with Proportional Sides are Similar
        
        Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar
        
        Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
        
        Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles
        
        Proposition: 6.09: Construction of Part of Line
        
        Proposition: 6.10: Construction of Similarly Cut Straight Line
        
        Proposition: 6.11: Construction of Segment in Squared Ratio
        
        Proposition: 6.12: Construction of Fourth Proportional Straight Line
        
        Proposition: 6.13: Construction of Mean Proportional
        
        Proposition: 6.14: Characterization of Congruent Parallelograms
        
        Proposition: 6.15: Characterization of Congruent Triangles
        
        Proposition: 6.16: Rectangles Contained by Proportional Straight Lines
        
        Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines
        
        Proposition: 6.18: Construction of Similar Polygon
        
        Proposition: 6.19: Ratio of Areas of Similar Triangles
        
        Proposition: 6.20: Similar Polygons are Composed of Similar Triangles
        
        Proposition: 6.21: Similarity of Polygons is Transitive
        
        Proposition: 6.22: Similar Figures on Proportional Straight Lines
        
        Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms
        
        Proposition: 6.24: Parallelograms About Diameter are Similar
        
        Proposition: 6.25: Construction of Figure Similar to One and Equal to Another
        
        Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter
        
        Proposition: 6.27: Similar Parallelogram on Half a Straight Line
        
        Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
        
        Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
        
        Proposition: 6.30: Construction of the Inverse Golden Section
        
        Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle
        
        Proposition: 6.32: Triangles with Two Sides Parallel and Equal
        
        Proposition: 6.33: Angles in Circles have Same Ratio as Arcs
      - Book 7 Elementary Number Theory (39)
        
        Proposition: 7.01: Sufficient Condition for Coprimality
        
        Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm
        
        Proposition: 7.03: Greatest Common Divisor of Three Numbers
        
        Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers
        
        Proposition: 7.05: Divisors Obey Distributive Law (Sum)
        
        Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum)
        
        Proposition: 7.07: Divisors Obey Distributive Law (Difference)
        
        Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference)
        
        Proposition: 7.09: Alternate Ratios of Equal Fractions
        
        Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions
        
        Proposition: 7.11: Proportional Numbers have Proportional Differences
        
        Proposition: 7.12: Ratios of Numbers is Distributive over Addition
        
        Proposition: 7.13: Proportional Numbers are Proportional Alternately
        
        Proposition: 7.14: Proportion of Numbers is Transitive
        
        Proposition: 7.15: Alternate Ratios of Multiples
        
        Proposition: 7.16: Natural Number Multiplication is Commutative
        
        Proposition: 7.17: Multiples of Ratios of Numbers
        
        Proposition: 7.18: Ratios of Multiples of Numbers
        
        Proposition: 7.19: Relation of Ratios to Products
        
        Proposition: 7.20: Ratios of Fractions in Lowest Terms
        
        Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms
        
        Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime
        
        Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other
        
        Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole
        
        Proposition: 7.25: Square of Co-prime Number is Co-prime
        
        Proposition: 7.26: Product of Co-prime Pairs is Co-prime
        
        Proposition: 7.27: Powers of Co-prime Numbers are Co-prime
        
        Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both
        
        Proposition: 7.29: Prime not Divisor implies Co-prime
        
        Proposition: 7.30: Euclidean Lemma
        
        Proposition: 7.31: Existence of Prime Divisors
        
        Proposition: 7.32: Natural Number is Prime or has Prime Factor
        
        Proposition: 7.33: Least Ratio of Numbers
        
        Proposition: 7.34: Existence of Least Common Multiple
        
        Proposition: 7.35: Least Common Multiple Divides Common Multiple
        
        Proposition: 7.36: Least Common Multiple of Three Numbers
        
        Proposition: 7.37: Integer Divided by Divisor is Integer
        
        Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer
        
        Proposition: 7.39: Least Number with Three Given Fractions
      - Book 8 Continued Proportion (27)
        
        Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms
        
        Proposition: 8.02: Construction of Geometric Progression in Lowest Terms
        
        Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes
        
        Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios
        
        Proposition: 8.05: Ratio of Products of Sides of Plane Numbers
        
        Proposition: 8.06: First Element of Geometric Progression not dividing Second
        
        Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second
        
        Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements
        
        Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers
        
        Proposition: Prop. 8.10: Product of Geometric Progressions from One
        
        Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional
        
        Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals
        
        Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression
        
        Proposition: Prop. 8.14: Number divides Number iff Square divides Square
        
        Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube
        
        Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square
        
        Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube
        
        Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional
        
        Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals
        
        Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane
        
        Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid
        
        Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square
        
        Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube
        
        Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square
        
        Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
        
        Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares
        
        Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes
      - Book 9 Number Theory Applications (35)
        
        Proposition: 9.35: Sum of Geometric Progression
        
        Proposition: 9.36: Theorem of Even Perfect Numbers (First Part)
        
        Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square
        
        Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers
        
        Proposition: Prop. 9.03: Square of Cube Number is Cube
        
        Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube
        
        Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube
        
        Proposition: Prop. 9.06: Number Squared making Cube is itself Cube
        
        Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number
        
        Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number
        
        Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number
        
        Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number
        
        Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements
        
        Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime
        
        Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime
        
        Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element
        
        Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional
        
        Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second
        
        Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers
        
        Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers
        
        Proposition: Prop. 9.20: Infinite Number of Primes
        
        Proposition: Prop. 9.21: Sum of Even Numbers is Even
        
        Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even
        
        Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd
        
        Proposition: Prop. 9.24: Even Number minus Even Number is Even
        
        Proposition: Prop. 9.25: Even Number minus Odd Number is Odd
        
        Proposition: Prop. 9.26: Odd Number minus Odd Number is Even
        
        Proposition: Prop. 9.27: Odd Number minus Even Number is Odd
        
        Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even
        
        Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd
        
        Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half
        
        Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double
        
        Proposition: Prop. 9.32: Power of Two is Even-Times Even Only
        
        Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd
        
        Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
      - Book 10 Incommensurable Magnitudes (115)
        
        Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number
        
        Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm
        
        Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes
        
        Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes
        
        Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes
        
        Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable
        
        Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio
        
        Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable
        
        Proposition: Prop. 10.009: Commensurability of Squares
        
        Proposition: Prop. 10.010: Construction of Incommensurable Lines
        
        Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes
        
        Proposition: Prop. 10.012: Commensurability is Transitive Relation
        
        Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
        
        Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines
        
        Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes
        
        Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
        
        Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation
        
        Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation
        
        Proposition: Prop. 10.019: Product of Rational Numbers is Rational
        
        Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational
        
        Proposition: Prop. 10.021: Medial is Irrational
        
        Proposition: Prop. 10.022: Square on Medial Straight Line
        
        Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial
        
        Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
        
        Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
        
        Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area
        
        Proposition: Prop. 10.027: Construction of Components of First Bimedial
        
        Proposition: Prop. 10.028: Construction of Components of Second Bimedial
        
        Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable
        
        Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable
        
        Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I
        
        Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II
        
        Proposition: Prop. 10.033: Construction of Components of Major
        
        Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area
        
        Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas
        
        Proposition: Prop. 10.036: Binomial is Irrational
        
        Proposition: Prop. 10.037: First Bimedial is Irrational
        
        Proposition: Prop. 10.038: Second Bimedial is Irrational
        
        Proposition: Prop. 10.039: Major is Irrational
        
        Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational
        
        Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational
        
        Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely
        
        Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely
        
        Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely
        
        Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely
        
        Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely
        
        Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely
        
        Proposition: Prop. 10.048: Construction of First Binomial Straight Line
        
        Proposition: Prop. 10.049: Construction of Second Binomial Straight Line
        
        Proposition: Prop. 10.050: Construction of Third Binomial Straight Line
        
        Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line
        
        Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line
        
        Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line
        
        Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial
        
        Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial
        
        Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial
        
        Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial
        
        Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial
        
        Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial
        
        Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line
        
        Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line
        
        Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
        
        Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
        
        Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major
        
        Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area
        
        Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas
        
        Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
        
        Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
        
        Proposition: Prop. 10.073: Apotome is Irrational
        
        Proposition: Prop. 10.074: First Apotome of Medial is Irrational
        
        Proposition: Prop. 10.075: Second Apotome of Medial is Irrational
        
        Proposition: Prop. 10.076: Minor is Irrational
        
        Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational
        
        Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational
        
        Proposition: Prop. 10.079: Construction of Apotome is Unique
        
        Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique
        
        Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique
        
        Proposition: Prop. 10.082: Construction of Minor is Unique
        
        Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique
        
        Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique
        
        Proposition: Prop. 10.085: Construction of First Apotome
        
        Proposition: Prop. 10.086: Construction of Second Apotome
        
        Proposition: Prop. 10.087: Construction of Third Apotome
        
        Proposition: Prop. 10.088: Construction of Fourth Apotome
        
        Proposition: Prop. 10.089: Construction of Fifth Apotome
        
        Proposition: Prop. 10.090: Construction of Sixth Apotome
        
        Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome
        
        Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome
        
        Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome
        
        Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome
        
        Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome
        
        Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome
        
        Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line
        
        Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line
        
        Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
        
        Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
        
        Proposition: Prop. 10.103: Straight Line Commensurable with Apotome
        
        Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line
        
        Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line
        
        Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area
        
        Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area
        
        Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted
        
        Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
        
        Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
        
        Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line
        
        Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line
        
        Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome
        
        Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
        
        Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines
      - Book 11 Elementary Stereometry (39)
        
        Proposition: 11.02: Two Intersecting Straight Lines are in One Plane
        
        Proposition: Prop. 11.01: Straight Line cannot be in Two Planes
        
        Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line
        
        Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
        
        Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane
        
        Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel
        
        Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane
        
        Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
        
        Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other
        
        Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
        
        Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane
        
        Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane
        
        Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique
        
        Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel
        
        Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel
        
        Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel
        
        Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes
        
        Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
        
        Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
        
        Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
        
        Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles
        
        Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
        
        Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles
        
        Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
        
        Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes
        
        Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle
        
        Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped
        
        Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
        
        Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
        
        Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
        
        Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
        
        Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases
        
        Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
        
        Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
        
        Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
        
        Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme
        
        Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
        
        Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
        
        Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base
      - Book 12 Proportional Stereometry (18)
        
        Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
        
        Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters
        
        Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
        
        Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
        
        Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases
        
        Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
        
        Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra
        
        Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides
        
        Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
        
        Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height
        
        Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
        
        Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
        
        Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
        
        Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
        
        Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
        
        Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
        
        Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres
        
        Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters
      - Book 13 Platonic Solids (18)
        
        Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
        
        Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
        
        Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
        
        Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
        
        Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
        
        Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
        
        Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal
        
        Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
        
        Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
        
        Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa
        
        Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
        
        Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
        
        Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere
        
        Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere
        
        Proposition: Prop. 13.15: Construction of Cube within Given Sphere
        
        Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere
        
        Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere
        
        Proposition: Prop. 13.18: There are only Five Platonic Solids
- Graph Theory (9)
  - Proposition: A Necessary Condition for a Graph to be Planar
  - Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar
  - Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II)
  - Proposition: Characterization of Cutvertices
  - Proposition: Chromatic Number and Maximum Vertex Degree
  - Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph
  - Proposition: Equivalent Definitions of Trees
  - Proposition: Relationship Between Planarity and Biconnectivity of Graphs
  - Proposition: Relationship Between Planarity and Connectivity of Graphs
- Knot Theory (1)
  - Proposition: Equivalent Knot Diagrams
- Logic (2)
  - Proposition: Associativity of Conjunction
  - Proposition: Associativity of Disjunction
- Number Systems Arithmetics (160)
  - Proposition: Abelian Partial Summation Method
  - Proposition: Absolute Value of Complex Conjugate
  - Proposition: Absolute Value of the Product of Complex Numbers
  - Proposition: Addition Of Natural Numbers
  - Proposition: Addition Of Natural Numbers Is Associative
  - Proposition: Addition Of Rational Numbers
  - Proposition: Addition Of Real Numbers Is Associative
  - Proposition: Addition Of Real Numbers Is Commutative
  - Proposition: Addition of Complex Numbers Is Associative
  - Proposition: Addition of Complex Numbers Is Commutative
  - Proposition: Addition of Integers
  - Proposition: Addition of Integers Is Associative
  - Proposition: Addition of Integers Is Cancellative
  - Proposition: Addition of Integers Is Commutative
  - Proposition: Addition of Natural Numbers Is Cancellative
  - Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities
  - Proposition: Addition of Natural Numbers Is Commutative
  - Proposition: Addition of Rational Cauchy Sequences
  - Proposition: Addition of Rational Cauchy Sequences Is Associative
  - Proposition: Addition of Rational Cauchy Sequences Is Cancellative
  - Proposition: Addition of Rational Cauchy Sequences Is Commutative
  - Proposition: Addition of Rational Numbers Is Associative
  - Proposition: Addition of Rational Numbers Is Cancellative
  - Proposition: Addition of Rational Numbers Is Commutative
  - Proposition: Addition of Real Numbers
  - Proposition: Addition of Real Numbers Is Cancellative
  - Proposition: Algebraic Structure Of Natural Numbers Together With Addition
  - Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication
  - Proposition: Algebraic Structure of Complex Numbers Together with Addition
  - Proposition: Algebraic Structure of Complex Numbers Together with Addition and Multiplication
  - Proposition: Algebraic Structure of Integers Together with Addition
  - Proposition: Algebraic Structure of Integers Together with Addition and Multiplication
  - Proposition: Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication
  - Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication
  - Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication
  - Proposition: Algebraic Structure of Rational Numbers Together with Addition
  - Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication
  - Proposition: Algebraic Structure of Real Numbers Together with Addition
  - Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication
  - Proposition: Alternating Sum of Binomial Coefficients
  - Proposition: Basic Rules of Manipulating Finite Sums
  - Proposition: Calculating with Complex Conjugates
  - Proposition: Comparing Natural Numbers Using the Concept of Addition
  - Proposition: Complex Numbers Cannot Be Ordered
  - Proposition: Complex Numbers are a Field Extension of Real Numbers
  - Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers
  - Proposition: Contraposition of Cancellative Law for Adding Integers
  - Proposition: Contraposition of Cancellative Law for Adding Rational Numbers
  - Proposition: Contraposition of Cancellative Law for Adding Real Numbers
  - Proposition: Contraposition of Cancellative Law for Multiplying Integers
  - Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers
  - Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers
  - Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers
  - Proposition: Definition of Integers
  - Proposition: Definition of Rational Numbers
  - Proposition: Definition of Real Numbers
  - Proposition: Discovery of Irrational Numbers
  - Proposition: Distributivity Law For Integers
  - Proposition: Distributivity Law For Natural Numbers
  - Proposition: Distributivity Law For Rational Cauchy Sequences
  - Proposition: Distributivity Law For Rational Numbers
  - Proposition: Distributivity Law For Real Numbers
  - Proposition: Distributivity Law for Complex Numbers
  - Proposition: Double Summation
  - Proposition: Equality of Two Ratios
  - Proposition: Every Natural Number Is Greater or Equal Zero
  - Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers
  - Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers)
  - Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers)
  - Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers)
  - Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers)
  - Proposition: Existence of Inverse Complex Numbers With Respect to Addition
  - Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication
  - Proposition: Existence of Inverse Integers With Respect to Addition
  - Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition
  - Proposition: Existence of Inverse Rational Numbers With Respect to Addition
  - Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication
  - Proposition: Existence of Inverse Real Numbers With Respect to Addition
  - Proposition: Existence of Inverse Real Numbers With Respect to Multiplication
  - Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences)
  - Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences)
  - Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)
  - Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers)
  - Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers)
  - Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers)
  - Proposition: Extracting the Real and the Imaginary Part of a Complex Number
  - Proposition: Geometric Sum
  - Proposition: Imaginary Unit
  - Proposition: Inequality of Natural Numbers and Their Successors
  - Proposition: Multiplication Of Rational Cauchy Sequences
  - Proposition: Multiplication Of Rational Numbers
  - Proposition: Multiplication Of Rational Numbers Is Cancellative
  - Proposition: Multiplication Of Rational Numbers Is Commutative
  - Proposition: Multiplication of Complex Numbers Is Associative
  - Proposition: Multiplication of Complex Numbers Is Commutative
  - Proposition: Multiplication of Complex Numbers Using Polar Coordinates
  - Proposition: Multiplication of Integers
  - Proposition: Multiplication of Integers Is Associative
  - Proposition: Multiplication of Integers Is Cancellative
  - Proposition: Multiplication of Integers Is Commutative
  - Proposition: Multiplication of Natural Numbers Is Associative
  - Proposition: Multiplication of Natural Numbers Is Cancellative
  - Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation
  - Proposition: Multiplication of Natural Numbers is Commutative
  - Proposition: Multiplication of Rational Cauchy Sequences Is Associative
  - Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative
  - Proposition: Multiplication of Rational Cauchy Sequences Is Commutative
  - Proposition: Multiplication of Rational Numbers Is Associative
  - Proposition: Multiplication of Real Numbers
  - Proposition: Multiplication of Real Numbers Is Associative
  - Proposition: Multiplication of Real Numbers Is Cancellative
  - Proposition: Multiplication of Real Numbers Is Commutative
  - Proposition: Multiplying Negative and Positive Integers
  - Proposition: Multiplying Negative and Positive Rational Numbers
  - Proposition: Multiplying Negative and Positive Real Numbers
  - Proposition: Order Relation for Integers is Strict Total
  - Proposition: Order Relation for Natural Numbers, Revised
  - Proposition: Order Relation for Rational Numbers is Strict Total
  - Proposition: Order Relation for Real Numbers is Strict and Total
  - Proposition: Polar Coordinates of a Complex Number
  - Proposition: Position of Minus Sign in Rational Numbers Representations
  - Proposition: Product of Two Ratios
  - Proposition: Product of Two Sums (Generalized Distributivity Rule)
  - Proposition: Ratio of Two Ratios
  - Proposition: Rational Cauchy Sequence Members Are Bounded
  - Proposition: Rule of Combining Different Sets of Indices
  - Proposition: Sum and Difference of Two Ratios
  - Proposition: Sum of Arithmetic Progression
  - Proposition: Sum of Binomial Coefficients
  - Proposition: Sum of Binomial Coefficients I
  - Proposition: Sum of Binomial Coefficients II
  - Proposition: Sum of Binomial Coefficients III
  - Proposition: Sum of Binomial Coefficients IV
  - Proposition: Sum of Consecutive Natural Numbers
  - Proposition: Sum of Consecutive Odd Numbers
  - Proposition: Sum of Cosines
  - Proposition: Sum of Cube Numbers
  - Proposition: Sum of Factorials (I)
  - Proposition: Sum of Geometric Progression
  - Proposition: Sum of Squares
  - Proposition: The General Perturbation Method
  - Proposition: Transitivity of the Order Relation of Natural Numbers
  - Proposition: Unique Solvability of `$ax=b$`
  - Proposition: Unique Solvability of `\(a+x=b\)`
  - Proposition: Uniqueness Of Natural One
  - Proposition: Uniqueness Of Predecessors Of Natural Numbers
  - Proposition: Uniqueness Of Rational One
  - Proposition: Uniqueness of Complex Zero
  - Proposition: Uniqueness of Integer One
  - Proposition: Uniqueness of Integer Zero
  - Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication
  - Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication
  - Proposition: Uniqueness of Natural Zero
  - Proposition: Uniqueness of Negative Numbers
  - Proposition: Uniqueness of Rational Zero
  - Proposition: Uniqueness of Real One
  - Proposition: Uniqueness of Real Zero
  - Proposition: Well-Ordering Principle of Natural Numbers
  - Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)`
  - Proposition: `\(-(x+y)=-x-y\)`
- Number Theory (58)
  - Proposition: A Linear Term for 1 Using Two Co-prime Coefficients
  - Proposition: A Necessary Condition for an Integer to be Prime
  - Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`
  - Proposition: All Solutions Given a Solution of an LDE With Two Variables
  - Proposition: Calculating the Number of Positive Divisors
  - Proposition: Calculating the Sum of Divisors
  - Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$`
  - Proposition: Cancellation of Congruences with General Factor
  - Proposition: Co-prime Primes
  - Proposition: Complete and Reduced Residue Systems (Revised)
  - Proposition: Congruence Classes
  - Proposition: Congruence Modulo a Divisor
  - Proposition: Congruences and Division with Quotient and Remainder
  - Proposition: Connection between Quotient, Remainder, Modulo and Floor Function
  - Proposition: Convergence of Alternating Harmonic Series
  - Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number
  - Proposition: Counting the Solutions of Diophantine Equations of Congruences
  - Proposition: Creation of Complete Residue Systems From Others
  - Proposition: Creation of Reduced Residue Systems From Others
  - Proposition: Diophantine Equations of Congruences
  - Proposition: Divergence of Harmonic Series
  - Proposition: Divisibility Laws
  - Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor
  - Proposition: Euler's Criterion For Quadratic Residues
  - Proposition: Even Perfect Numbers
  - Proposition: Every Integer Is Either Even or Odd
  - Proposition: Existence and Number of Solutions of Congruence With One Variable
  - Proposition: Existence of Prime Divisors
  - Proposition: Existence of Solutions of an LDE With More Variables
  - Proposition: Explicit Formula for the Euler Function
  - Proposition: Factorization of Greatest Common Divisor and Least Common Multiple
  - Proposition: Finite Number of Divisors
  - Proposition: Floor Function and Division with Quotient and Remainder
  - Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor
  - Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers
  - Proposition: Greatest Common Divisor
  - Proposition: Greatest Common Divisor of More Than Two Numbers
  - Proposition: Greatest Common Divisors Of Integers and Prime Numbers
  - Proposition: Least Common Multiple
  - Proposition: Least Common Multiple of More Than Two Numbers
  - Proposition: Legendre Symbols of Equal Residues
  - Proposition: Multiplication of Congruences with a Positive Factor
  - Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$`
  - Proposition: Multiplicativity of the Legendre Symbol
  - Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors
  - Proposition: Natural Numbers and Products of Prime Numbers
  - Proposition: Number of Multiples of a Given Number Less Than Another Number
  - Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime
  - Proposition: Numbers Being the Product of Their Divisors
  - Proposition: Product of Two Even Numbers
  - Proposition: Product of Two Odd Numbers
  - Proposition: Product of an Even and an Odd Number
  - Proposition: Properties of Floors and Ceilings
  - Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple
  - Proposition: Sign of Divisors of Integers
  - Proposition: Sum of Euler Function
  - Proposition: Sum of Möbius Function Over Divisors
  - Proposition: Wilson's Condition for an Integer to be Prime
- Probability Theory And Statistics (14)
  - Proposition: Binomial Distribution
  - Proposition: Characterization of Independent Events
  - Proposition: Characterization of Independent Events II
  - Proposition: Geometric Distribution
  - Proposition: Monotonically Increasing Property of Probability Distributions
  - Proposition: Multinomial Distribution
  - Proposition: Probability of Event Difference
  - Proposition: Probability of Event Union
  - Proposition: Probability of Included Event
  - Proposition: Probability of Joint Events
  - Proposition: Probability of the Complement Event
  - Proposition: Replacing Mutually Independent Events by Their Complements
  - Proposition: Urn Model With Replacement
  - Proposition: Urn Model Without Replacement
- Set Theory (39)
  - Proposition: Cardinal Number
  - Proposition: Cardinals of a Set and Its Power Set
  - Proposition: Characterization of Bijective Functions
  - Proposition: Composition of Bijective Functions is Bijective
  - Proposition: Composition of Functions is Associative
  - Proposition: Composition of Injective Functions is Injective
  - Proposition: Composition of Surjective Functions is Surjective
  - Proposition: Contained Relation is a Strict Order
  - Proposition: Counting the Set's Elements Using Its Partition
  - Proposition: De Morgan's Laws (Sets)
  - Proposition: Distributivity Laws For Sets
  - Proposition: Equivalent Notions of Ordinals
  - Proposition: Finite Chains are Well-ordered
  - Proposition: Functions Constitute Equivalence Relations
  - Proposition: Injective, Surjective and Bijective Compositions
  - Proposition: Intersection of a Set With Another Set is Subset of This Set
  - Proposition: More Characterizations of Finite Sets
  - Proposition: Ordinals Are Downward Closed
  - Proposition: Partial Orders are Extensional
  - Proposition: Rational Numbers are Countable
  - Proposition: Real Numbers are Uncountable
  - Proposition: Set Intersection is Associative
  - Proposition: Set Intersection is Commutative
  - Proposition: Set Union is Associative
  - Proposition: Set Union is Commutative
  - Proposition: Set-Theoretical Meaning of Ordered Tuples
  - Proposition: Sets and Their Complements
  - Proposition: Sets are Subsets of Their Union
  - Proposition: Strict Orders are Extensional
  - Proposition: Subset of a Countable Set is Countable
  - Proposition: Subsets of Finite Sets
  - Proposition: The Contained Relation is Extensional
  - Proposition: The Equality of Sets Is an Equivalence Relation
  - Proposition: The Inverse Of a Composition
  - Proposition: Transitive Recursion
  - Proposition: Uncountable and Countable Subsets of Natural Numbers
  - Proposition: Union of Countably Many Countable Sets
  - Proposition: Well-ordered Sets are Chains
  - Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice
- Theoretical Computer Science (1)
  - Complexity Theory (1)
    - Proposition: Calculation Rules for the Big O Notation
- Theoretical Physics (2)
  - Special Relativity (2)
    - Proposition: Construction of a Light Clock
    - Proposition: Time Dilation, Lorentz Factor
- Topology (28)
  - Proposition: A Necessary Condition of a Neighborhood to be Open
  - Proposition: Alternative Characterization of Topological Spaces
  - Proposition: Bijective Open Functions
  - Proposition: Characterization of `$T_1$` Spaces
  - Proposition: Characterization of `$T_2$` Spaces
  - Proposition: Clopen Sets and Boundaries
  - Proposition: Construction of Topological Spaces Using a Subbasis
  - Proposition: Continuity of Compositions of Functions
  - Proposition: Distance in Normed Vector Spaces
  - Proposition: Equivalent Notions of Continuous Functions
  - Proposition: Equivalent Notions of Homeomorphisms
  - Proposition: Filter Base
  - Proposition: How the Boundary Changes the Property of a Set of Being Open
  - Proposition: Inheritance of the `$T_1$` Property
  - Proposition: Inheritance of the `$T_2$` Property
  - Proposition: Integral p-Norm
  - Proposition: Isometry is Injective
  - Proposition: Maximum Norm as a Limit of p-Norms
  - Proposition: Metric Spaces and Empty Sets are Clopen
  - Proposition: Metric Spaces are Hausdorff Spaces
  - Proposition: Modulus of Continuity is Continuous
  - Proposition: Modulus of Continuity is Monotonically Increasing
  - Proposition: Modulus of Continuity is Subadditive
  - Proposition: Perfect Sets vs. Derived Sets
  - Proposition: Properties of the Set of All Neighborhoods of a Point
  - Proposition: Uniqueness of the Limit of a Sequence
  - Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity
  - Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm
Corollaries (147)
Proofs (1446)