- 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)

- Euclidean Geometry (7)
- Logic (1)
- Axiom: Bivalence of Truth

- Number Systems Arithmetics (2)
- Axiom: Archimedean Axiom
- Axiom: Peano Axioms

- Probability Theory And Statistics (3)
- 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)

- Topology (2)
- Axiom: Filter
- Axiom: Separation Axioms

- Algebra (5)
- 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

- Book 1 Plane Geometry (35)

- 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

- Analytic Geometry (4)
- 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: MÃ¶bius 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

- Complexity Theory (1)
- 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

- Classical Physics (2)
- 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

- Algebra (138)
- 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)
- Euclidean Geometry (1)
- Elements Euclid (1)
- Book 9 Number Theory Applications (1)

- Elements Euclid (1)

- 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: MÃ¶bius 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: SchrÃ¶der-Bernstein Theorem
- Theorem: Trichotomy of Ordinals

- Theoretical Computer Science (5)
- Computability (2)
- Formal Languages (3)

- Theoretical Physics (3)
- Classical Physics (3)
- Theorem: First Law of Planetary Motion
- Theorem: Second Law of Planetary Motion
- Theorem: Third Law of Planetary Motion

- Classical Physics (3)
- Topology (2)

- Algebra (12)
- 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)
- Geometry (16)
- Analytic Geometry (1)
- 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)
- Book 12 Proportional Stereometry (2)
- Book 13 Platonic Solids (3)

- Book 10 Incommensurable Magnitudes (9)

- Elements Euclid (15)

- Graph Theory (11)
- Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph
- Lemma: Coloring of Trees
- Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree
- Lemma: Handshaking Lemma for Finite Digraphs
- Lemma: Handshaking Lemma for Finite Graphs
- Lemma: Handshaking Lemma for Planar Graphs
- Lemma: Lower Bound of Leaves in a Tree
- Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree
- Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices
- Lemma: Splitting a Graph with Even Degree Vertices into Cycles
- Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?

- 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)
- Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are Rational Cauchy Sequences
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are a Subgroup of Rational Cauchy Sequences With Respect To Addition
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are an Ideal Of the Ring of Rational Cauchy Sequences
- Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`
- Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition
- Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication
- Lemma: Unit Ring of All Rational Cauchy Sequences

- Number Theory (11)
- Lemma: Coprimality and Congruence Classes
- Lemma: Division with Quotient and Remainder
- Lemma: Gaussian Lemma (Number Theory)
- Lemma: Generalized Euclidean Lemma
- Lemma: MÃ¶bius 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 MÃ¶bius Function Over Divisors With Division
- Lemma: Sums of Floors
- Lemma: Upper Bound of Harmonic Series Times MÃ¶bius 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)

- Topology (3)

- Algebra (22)
- 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: Cauchyâ€“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: HÃ¶lder's Inequality
- Proposition: HÃ¶lder'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)
- 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

- Book 1 Plane Geometry (48)

- 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 MÃ¶bius 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

- Complexity Theory (1)
- Theoretical Physics (2)
- Special Relativity (2)
- Proposition: Construction of a Light Clock
- Proposition: Time Dilation, Lorentz Factor

- Special Relativity (2)
- 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

- Algebra (32)
- Corollaries (147)
- Algebra (8)
- Corollary: Abelian Group of Vectors Under Addition (related to Proposition: Abelian Group of Matrices Under Addition)
- Corollary: Barycentric Coordinates, Barycenter (related to Definition: Affine Basis, Affine Coordinate System)
- Corollary: General Associative Law (related to Definition: Associativity)
- Corollary: General Commutative Law (related to Definition: Commutativity)
- Corollary: Intersection of Convex Affine Sets (related to Definition: Convex Affine Set)
- Corollary: Properties of a Real Scalar Product (related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case))
- Corollary: Rules for Exponentiation in a Group (related to Definition: Exponentiation in a Group)
- Corollary: Solutions of a Linear Equation with many Unknowns (related to Definition: Linear Equations with many Unknowns)

- Analysis (33)
- Corollary: (Real) Exponential Function Is Always Positive (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
- Corollary: All Uniformly Continuous Functions are Continuous (related to Definition: Uniformly Continuous Functions (Real Case))
- Corollary: All Zeros of Cosine and Sine (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Arguments for which Cosine and Sine are Equal to Each Other (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Closed Real Intervals Are Compact (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
- Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
- Corollary: Continuous Real Functions on Closed Intervals are Bounded (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
- Corollary: Convergence of Complex Conjugate Sequence (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
- Corollary: Cosine and Sine are Periodic Functions (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Derivative of a Constant Function (related to Definition: Constant Function Real Case)
- Corollary: Derivative of a Linear Function `\(ax+b\)` (related to Definition: Linear Function)
- Corollary: Differentiable Functions are Continuous (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
- Corollary: Estimating the Growth of a Function with its Derivative (related to Theorem: Darboux's Theorem)
- Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments (related to Definition: Continuous Functions at Single Real Numbers)
- Corollary: Exponential Function Is Non-Negative (Real Case) (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Exponential Function Is Strictly Monotonically Increasing (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Exponential Function and the Euler Constant (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point (related to Definition: Continuous Functions at Single Real Numbers)
- Corollary: Limit of N-th Roots (related to Proposition: Rational Powers of Positive Numbers)
- Corollary: Monotony Criterion for Absolute Series (related to Proposition: Monotony Criterion)
- Corollary: More Insight to Euler's Identity (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Negative Cosine and Sine vs Shifting the Argument (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Non-Cauchy Sequences are Not Convergent (related to Theorem: Completeness Principle for Real Numbers)
- Corollary: Real Numbers Can Be Approximated by Rational Numbers (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
- Corollary: Real Polynomials of Odd Degree Have at Least One Real Root (related to Proposition: Limits of Polynomials at Infinity)
- Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Complex Exponential Function)
- Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function of General Base)
- Corollary: Reciprocity of Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Representing Real Cosine by Complex Exponential Function (related to Definition: Cosine of a Real Variable)
- Corollary: Representing Real Sine by Complex Exponential Function (related to Definition: Sine of a Real Variable)
- Corollary: Sufficient Condition for a Function to be Constant (related to Corollary: Estimating the Growth of a Function with its Derivative)
- Corollary: Taylor's Formula for Polynomials (related to Theorem: Taylor's Formula)
- Corollary: Value of Zero to the Power of X (related to Proposition: Limits of General Powers)

- Combinatorics (1)
- Geometry (45)
- Euclidean Geometry (45)
- Elements Euclid (45)
- Book 1 Plane Geometry (20)
- Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II)
- Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III)
- Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I)
- Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines)
- Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)

- Book 3 Circles (2)
- Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle)
- Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)

- Book 4 Inscription And Circumscription (1)
- Book 5 Proportion (2)
- Book 6 Similar Figures (3)
- Corollary: 6.08: Geometric Mean Theorem (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
- Corollary: 6.19: Ratio of Areas of Similar Triangles (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
- Corollary: 6.20: Similar Polygons are Composed of Similar Triangles (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)

- Book 7 Elementary Number Theory (1)
- Book 8 Continued Proportion (1)
- Book 9 Number Theory Applications (1)
- Book 10 Incommensurable Magnitudes (7)
- Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Corollary: Cor. 10.009: Commensurability of Squares (related to Proposition: Prop. 10.009: Commensurability of Squares)
- Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
- Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
- Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)

- Book 11 Elementary Stereometry (2)
- Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
- Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)

- Book 12 Proportional Stereometry (3)
- Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
- Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)

- Book 13 Platonic Solids (2)
- Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
- Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)

- Book 1 Plane Geometry (20)

- Elements Euclid (45)

- Euclidean Geometry (45)
- Graph Theory (5)
- Corollary: Bounds for the Minimal Tree Decomposability (related to Definition: Minimal Tree Decomposability)
- Corollary: Even Number of Vertices with an Odd Degree in Finite Digraphs (related to Lemma: Handshaking Lemma for Finite Digraphs)
- Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs (related to Lemma: Handshaking Lemma for Finite Graphs)
- Corollary: Number of Vertices, Edges, and Faces of a Dual Graph (related to Definition: Dual Planar Graph)
- Corollary: Planarity of Subdivisions (related to Definition: Subdivision of a Graph)

- Logic (5)
- Corollary: Algebraic Structure of Strings over an Alphabet (related to Definition: Strings (words) over an Alphabet)
- Corollary: All Boolean Functions Can Be Built Using Conjunction, Disjunction, and Negation (related to Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms)
- Corollary: Commutativity of Conjunction (related to Definition: Conjunction)
- Corollary: Commutativity of Disjunction (related to Definition: Disjunction)
- Corollary: Commutativity of Equivalence (related to Definition: Equivalence)

- Number Systems Arithmetics (22)
- Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero. (related to Corollary: `\(0x=0\)`)
- Corollary: Contraposition of Cancellative Law for Adding Natural Numbers (related to Proposition: Addition of Natural Numbers Is Cancellative)
- Corollary: Existence of Arbitrarily Small Positive Rational Numbers (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Arbitrarily Small Powers (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle) (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) (related to Definition: Multiplication of Natural Numbers)
- Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) (related to Proposition: Addition Of Natural Numbers)
- Corollary: Existence of Powers Exceeding Any Positive Constant (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Unique Integers Exceeding Real Numbers (related to Axiom: Archimedean Axiom)
- Corollary: General Associative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Associative)
- Corollary: General Commutative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Commutative)
- Corollary: Order Relation for Natural Numbers is Strict Total (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
- Corollary: Properties of the Absolute Value (related to Definition: Absolute Value of Real Numbers (Modulus))
- Corollary: Rules of Calculations with Inequalities (related to Definition: Order Relation of Real Numbers)
- Corollary: The absolute value makes the set of rational numbers a metric space. (related to Definition: Absolute Value of Rational Numbers)
- Corollary: `\((-x)(-y)=xy\)` (related to Corollary: `\((-x)y=-(xy)\)`)
- Corollary: `\((-x)y=-(xy)\)` (related to Proposition: Uniqueness of Negative Numbers)
- Corollary: `\((x^{-1})^{-1}=x\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
- Corollary: `\(-(-x)=x\)` (related to Proposition: Uniqueness of Negative Numbers)
- Corollary: `\(-0=0\)` (related to Proposition: Uniqueness of Negative Numbers)
- Corollary: `\(0x=0\)` (related to Proposition: Distributivity Law For Real Numbers)
- Corollary: `\(1^{-1}=1\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)

- Number Theory (6)
- Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions (related to Proposition: Diophantine Equations of Congruences)
- Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor)
- Corollary: Primality of the Smallest Non-Trivial Divisor (related to Proposition: Existence of Prime Divisors)
- Corollary: Prime Dividing Product of Primes Implies Prime Divisor (related to Lemma: Generalized Euclidean Lemma)
- Corollary: Simple Conclusions For Multiplicative Functions (related to Definition: Multiplicative Functions)
- Corollary: Sums, Products, and Powers Of Congruences (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`)

- Probability Theory And Statistics (2)
- Set Theory (16)
- Corollary: Cartesian Products of Countable Sets Is Countable (related to Proposition: Union of Countably Many Countable Sets)
- Corollary: Circular References Of Self-Contained Sets Are Forbidden (related to Axiom: Axiom of Foundation)
- Corollary: Equality of Sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Irrational Numbers are Uncountable (related to Proposition: Real Numbers are Uncountable)
- Corollary: Justification of Power Set (related to Axiom: Axiom of Power Set)
- Corollary: Justification of Set Intersection (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Justification of Set Union (related to Axiom: Axiom of Union)
- Corollary: Justification of Subsets and Supersets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Justification of the Difference (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Justification of the Set-Builder Notation (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets (related to Axiom: Axiom of Infinity)
- Corollary: Properties of Transitive Sets (related to Definition: Transitive Set)
- Corollary: Set Difference and Set Complement are the Same Concepts (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Strictly, Well-ordered Sets and Transitive Sets (related to Theorem: Mostowski's Theorem)
- Corollary: There is no set of all sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Uniqueness of the Empty Set (related to Axiom: Axiom of Empty Set)

- Theoretical Computer Science (2)
- Computability (1)
- Formal Languages (1)

- Topology (2)

- Algebra (8)
- Proofs (1446)
- Algebra (73)
- Proof: (related to Corollary: Abelian Group of Vectors Under Addition)
- Proof: (related to Corollary: Barycentric Coordinates, Barycenter)
- Proof: (related to Corollary: General Associative Law)
- Proof: (related to Corollary: General Commutative Law)
- Proof: (related to Corollary: Intersection of Convex Affine Sets)
- Proof: (related to Corollary: Properties of a Real Scalar Product)
- Proof: (related to Corollary: Rules for Exponentiation in a Group)
- Proof: (related to Corollary: Solutions of a Linear Equation with many Unknowns)
- Proof: (related to Lemma: Any Positive Characteristic Is a Prime Number)
- Proof: (related to Lemma: Cyclic Groups are Abelian)
- Proof: (related to Lemma: Divisibility of Principal Ideals)
- Proof: (related to Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE)
- Proof: (related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
- Proof: (related to Lemma: Factor Groups)
- Proof: (related to Lemma: Factor Rings, Generalization of Congruence Classes)
- Proof: (related to Lemma: Fiber of Maximal Ideals)
- Proof: (related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
- Proof: (related to Lemma: Fiber of Prime Ideals)
- Proof: (related to Lemma: Greatest Common Divisor and Least Common Multiple of Ideals)
- Proof: (related to Lemma: Group Homomorphisms and Normal Subgroups)
- Proof: (related to Lemma: Kernel and Image of Group Homomorphism)
- Proof: (related to Lemma: Kernel and Image of a Group Homomorphism are Subgroups)
- Proof: (related to Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring)
- Proof: (related to Lemma: Prime Ideals of Multiplicative Systems in Integral Domains)
- Proof: (related to Lemma: Subgroups and Their Cosets are Equipotent)
- Proof: (related to Lemma: Subgroups of Cyclic Groups)
- Proof: (related to Lemma: Uniqueness Lemma of a Finite Basis)
- Proof: (related to Proposition: A Field with an Absolute Value is a Metric Space)
- Proof: (related to Proposition: Abelian Group of Matrices Under Addition)
- Proof: (related to Proposition: Additive Subgroups of Integers)
- Proof: (related to Proposition: Cancellation Law)
- Proof: (related to Proposition: Characterization of Dependent Absolute Values)
- Proof: (related to Proposition: Characterization of Non-Archimedean Absolute Values)
- Proof: (related to Proposition: Criteria for Subgroups)
- Proof: (related to Proposition: Finite Order of an Element Equals Order Of Generated Group)
- Proof: (related to Proposition: Group Homomorphisms with Cyclic Groups)
- Proof: (related to Proposition: Group of Units)
- Proof: (related to Proposition: In a Field, `$0$` Is Unequal `$1$`)
- Proof: (related to Proposition: Open and Closed Subsets of a Zariski Topology)
- Proof: (related to Proposition: Properties of Cosets)
- Proof: (related to Proposition: Properties of a Complex Scalar Product)
- Proof: (related to Proposition: Properties of a Group Homomorphism)
- Proof: (related to Proposition: Quadratic Formula)
- Proof: (related to Proposition: Quotient Space)
- Proof: (related to Proposition: Simple Calculations Rules in a Group)
- Proof: (related to Proposition: Simple Consequences from the Definition of a Vector Space)
- Proof: (related to Proposition: Spectrum Function of Commutative Rings)
- Proof: (related to Proposition: Square of a Non-Zero Element is Positive in Ordered Fields)
- Proof: (related to Proposition: Subgroups of Finite Cyclic Groups)
- Proof: (related to Proposition: Subset of Powers is a Submonoid)
- Proof: (related to Proposition: Unique Solvability of `$a\ast x=b$` in Groups)
- Proof: (related to Proposition: `$0$` Is Less Than `$1$` In Ordered Fields)
- Proof: (related to Theorem: Classification of Cyclic Groups)
- Proof: (related to Theorem: Classification of Finite Groups with the Order of a Prime Number)
- Proof: (related to Theorem: Connection between Rings, Ideals, and Fields)
- Proof: (related to Theorem: Construction of Fields from Integral Domains)
- Proof: (related to Theorem: Construction of Groups from Commutative and Cancellative Semigroups)
- Proof: (related to Theorem: Finite Basis Theorem)
- Proof: (related to Theorem: Finite Integral Domains are Fields)
- Proof: (related to Theorem: First Isomorphism Theorem for Groups)
- Proof: (related to Theorem: Order of Cyclic Group (Fermat's Little Theorem))
- Proof: (related to Theorem: Order of Subgroup Divides Order of Finite Group)
- Proof: (related to Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs)
- Proof: By Induction (related to Lemma: A Criterion for Associates)
- Proof: By Induction (related to Lemma: Continuants and Convergents)
- Proof: By Induction (related to Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations)
- Proof: By Induction (related to Proposition: Criterions for Equality of Principal Ideals)
- Proof: By Induction (related to Proposition: Generalization of Cancellative Multiplication of Integers)
- Proof: By Induction (related to Proposition: Principal Ideal Generated by A Unit)
- Proof: By Induction (related to Proposition: Principal Ideals being Maximal Ideals)
- Proof: By Induction (related to Proposition: Principal Ideals being Prime Ideals)
- Proof: Conformity (related to Proposition: Uniqueness of Inverse Elements)
- Proof: Zero (Absorbing, Annihilating) Element, Left Zero, Right Zero (related to Proposition: Uniqueness of the Neutral Element)

- Analysis (299)
- Proof: (related to Corollary: (Real) Exponential Function Is Always Positive)
- Proof: (related to Corollary: All Uniformly Continuous Functions are Continuous)
- Proof: (related to Corollary: All Zeros of Cosine and Sine)
- Proof: (related to Corollary: Arguments for which Cosine and Sine are Equal to Each Other)
- Proof: (related to Corollary: Closed Real Intervals Are Compact)
- Proof: (related to Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded)
- Proof: (related to Corollary: Continuous Real Functions on Closed Intervals are Bounded)
- Proof: (related to Corollary: Convergence of Complex Conjugate Sequence)
- Proof: (related to Corollary: Cosine and Sine are Periodic Functions)
- Proof: (related to Corollary: Derivative of a Constant Function)
- Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
- Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
- Proof: (related to Corollary: Differentiable Functions are Continuous)
- Proof: (related to Corollary: Estimating the Growth of a Function with its Derivative)
- Proof: (related to Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments)
- Proof: (related to Corollary: Exponential Function Is Non-Negative (Real Case))
- Proof: (related to Corollary: Exponential Function Is Strictly Monotonically Increasing)
- Proof: (related to Corollary: Exponential Function and the Euler Constant)
- Proof: (related to Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point)
- Proof: (related to Corollary: Limit of N-th Roots)
- Proof: (related to Corollary: Monotony Criterion for Absolute Series)
- Proof: (related to Corollary: More Insight to Euler's Identity)
- Proof: (related to Corollary: Negative Cosine and Sine vs Shifting the Argument)
- Proof: (related to Corollary: Non-Cauchy Sequences are Not Convergent)
- Proof: (related to Corollary: Real Numbers Can Be Approximated by Rational Numbers)
- Proof: (related to Corollary: Real Polynomials of Odd Degree Have at Least One Real Root)
- Proof: (related to Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property)
- Proof: (related to Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property)
- Proof: (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
- Proof: (related to Corollary: Representing Real Cosine by Complex Exponential Function)
- Proof: (related to Corollary: Representing Real Sine by Complex Exponential Function)
- Proof: (related to Corollary: Sufficient Condition for a Function to be Constant)
- Proof: (related to Corollary: Taylor's Formula for Polynomials)
- Proof: (related to Corollary: Value of Zero to the Power of X)
- Proof: (related to Lemma: Abel's Lemma for Testing Convergence)
- Proof: (related to Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals)
- Proof: (related to Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions)
- Proof: (related to Lemma: Convergence Test for Telescoping Series)
- Proof: (related to Lemma: Decreasing Sequence of Suprema of Extended Real Numbers)
- Proof: (related to Lemma: Euler's Identity)
- Proof: (related to Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point)
- Proof: (related to Lemma: Increasing Sequence of Infima of Extended Real Numbers)
- Proof: (related to Lemma: Invertible Functions on Real Intervals)
- Proof: (related to Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine)
- Proof: (related to Lemma: Sum of Roots Of Unity in Complete Residue Systems)
- Proof: (related to Lemma: Trapezoid Rule)
- Proof: (related to Lemma: Unit Circle)
- Proof: (related to Lemma: Upper Bound for the Product of General Powers)
- Proof: (related to Proposition: A General Criterion for the Convergence of Infinite Complex Series)
- Proof: (related to Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions)
- Proof: (related to Proposition: Abel's Test)
- Proof: (related to Proposition: Additivity Theorem of Tangent)
- Proof: (related to Proposition: Additivity Theorems of Cosine and Sine)
- Proof: (related to Proposition: Approximation of Functions by Taylor's Formula)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Difference)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Division)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Product)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Sums)
- Proof: (related to Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule)
- Proof: (related to Proposition: Bounds for Partial Sums of Exponential Series)
- Proof: (related to Proposition: Calculation Rules for General Powers)
- Proof: (related to Proposition: Calculations with Uniformly Convergent Functions)
- Proof: (related to Proposition: Cauchy Condensation Criterion)
- Proof: (related to Proposition: Cauchy Criterion)
- Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Complex Series)
- Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Series)
- Proof: (related to Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent)
- Proof: (related to Proposition: Cauchy-Schwarz Inequality for Integral p-norms)
- Proof: (related to Proposition: Cauchy-Schwarz Test)
- Proof: (related to Proposition: Cauchyâ€“Schwarz Inequality)
- Proof: (related to Proposition: Chain Rule)
- Proof: (related to Proposition: Characterization of Monotonic Functions via Derivatives)
- Proof: (related to Proposition: Closed Formula for the Maximum and Minimum of Two Numbers)
- Proof: (related to Proposition: Closed Subsets of Compact Sets are Compact)
- Proof: (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
- Proof: (related to Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum)
- Proof: (related to Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed)
- Proof: (related to Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth)
- Proof: (related to Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences)
- Proof: (related to Proposition: Complex Conjugate of Complex Exponential Function)
- Proof: (related to Proposition: Complex Convergent Sequences are Bounded)
- Proof: (related to Proposition: Complex Exponential Function)
- Proof: (related to Proposition: Composition of Continuous Functions at a Single Point)
- Proof: (related to Proposition: Compositions of Continuous Functions on a Whole Domain)
- Proof: (related to Proposition: Continuity of Complex Exponential Function)
- Proof: (related to Proposition: Continuity of Cosine and Sine)
- Proof: (related to Proposition: Continuity of Exponential Function of General Base)
- Proof: (related to Proposition: Continuity of Exponential Function)
- Proof: (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
- Proof: (related to Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable)
- Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity)
- Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero)
- Proof: (related to Proposition: Convergence Behavior of the Sequence `\((b^n)\)`)
- Proof: (related to Proposition: Convergence Behaviour of Absolutely Convergent Series)
- Proof: (related to Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero)
- Proof: (related to Proposition: Convergent Complex Sequences Are Bounded)
- Proof: (related to Proposition: Convergent Complex Sequences Are Cauchy Sequences)
- Proof: (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
- Proof: (related to Proposition: Convergent Real Sequences Are Cauchy Sequences)
- Proof: (related to Proposition: Convergent Real Sequences are Bounded)
- Proof: (related to Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space)
- Proof: (related to Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space)
- Proof: (related to Proposition: Convergent Sequences are Bounded)
- Proof: (related to Proposition: Convex Functions on Open Intervals are Continuous)
- Proof: (related to Proposition: Convexity and Concaveness Test)
- Proof: (related to Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm)
- Proof: (related to Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`)
- Proof: (related to Proposition: Derivative of Cosine)
- Proof: (related to Proposition: Derivative of General Powers of Positive Numbers)
- Proof: (related to Proposition: Derivative of Sine)
- Proof: (related to Proposition: Derivative of Tangent)
- Proof: (related to Proposition: Derivative of an Invertible Function on Real Invervals)
- Proof: (related to Proposition: Derivative of the Exponential Function)
- Proof: (related to Proposition: Derivative of the Inverse Sine)
- Proof: (related to Proposition: Derivative of the Inverse Tangent)
- Proof: (related to Proposition: Derivative of the Natural Logarithm)
- Proof: (related to Proposition: Derivative of the Reciprocal Function)
- Proof: (related to Proposition: Derivative of the n-th Power Function)
- Proof: (related to Proposition: Derivatives of Even and Odd Functions)
- Proof: (related to Proposition: Difference of Convergent Complex Sequences)
- Proof: (related to Proposition: Difference of Convergent Real Sequences)
- Proof: (related to Proposition: Difference of Convergent Real Series)
- Proof: (related to Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine)
- Proof: (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
- Proof: (related to Proposition: Differential Equation of the Exponential Function)
- Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Complex Series)
- Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Series)
- Proof: (related to Proposition: Direct Comparison Test For Divergent Series)
- Proof: (related to Proposition: Dirichlet's Test)
- Proof: (related to Proposition: Estimate for the Remainder Term of Complex Exponential Function)
- Proof: (related to Proposition: Estimate for the Remainder Term of Exponential Function)
- Proof: (related to Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine)
- Proof: (related to Proposition: Euler's Formula)
- Proof: (related to Proposition: Eveness (Oddness) of Polynomials)
- Proof: (related to Proposition: Eveness of the Cosine of a Real Variable)
- Proof: (related to Proposition: Exponential Function of General Base With Integer Exponents)
- Proof: (related to Proposition: Exponential Function)
- Proof: (related to Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals)
- Proof: (related to Proposition: Functional Equation of the Complex Exponential Function)
- Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base (Revised))
- Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base)
- Proof: (related to Proposition: Functional Equation of the Exponential Function)
- Proof: (related to Proposition: Functional Equation of the Natural Logarithm)
- Proof: (related to Proposition: Gamma Function Interpolates the Factorial)
- Proof: (related to Proposition: Gamma Function)
- Proof: (related to Proposition: General Powers of Positive Numbers)
- Proof: (related to Proposition: Generalized Product Rule)
- Proof: (related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)
- Proof: (related to Proposition: How Convergence Preserves the Order Relation of Sequence Members)
- Proof: (related to Proposition: HÃ¶lder's Inequality for Integral p-norms)
- Proof: (related to Proposition: HÃ¶lder's Inequality)
- Proof: (related to Proposition: Identity Function is Continuous)
- Proof: (related to Proposition: Image of a Compact Set Under a Continuous Function)
- Proof: (related to Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials)
- Proof: (related to Proposition: Infinite Geometric Series)
- Proof: (related to Proposition: Infinite Series for Cosine and Sine)
- Proof: (related to Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function)
- Proof: (related to Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function)
- Proof: (related to Proposition: Integral Test for Convergence)
- Proof: (related to Proposition: Integral of Cosine)
- Proof: (related to Proposition: Integral of General Powers)
- Proof: (related to Proposition: Integral of Inverse Sine)
- Proof: (related to Proposition: Integral of Sine)
- Proof: (related to Proposition: Integral of the Exponential Function)
- Proof: (related to Proposition: Integral of the Inverse Tangent)
- Proof: (related to Proposition: Integral of the Natural Logarithm)
- Proof: (related to Proposition: Integral of the Reciprocal Function)
- Proof: (related to Proposition: Inverse Cosine of a Real Variable)
- Proof: (related to Proposition: Inverse Hyperbolic Cosine)
- Proof: (related to Proposition: Inverse Hyperbolic Sine)
- Proof: (related to Proposition: Inverse Sine of a Real Variable)
- Proof: (related to Proposition: Inverse Tangent and Complex Exponential Function)
- Proof: (related to Proposition: Inverse Tangent of a Real Variable)
- Proof: (related to Proposition: Legendre Polynomials and Legendre Differential Equations)
- Proof: (related to Proposition: Leibniz Criterion for Alternating Series)
- Proof: (related to Proposition: Limit Comparizon Test)
- Proof: (related to Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence)
- Proof: (related to Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence)
- Proof: (related to Proposition: Limit Test for Roots or Ratios)
- Proof: (related to Proposition: Limit of 1/n)
- Proof: (related to Proposition: Limit of Exponential Growth as Compared to Polynomial Growth)
- Proof: (related to Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth)
- Proof: (related to Proposition: Limit of Nested Real Intervals)
- Proof: (related to Proposition: Limit of Nth Root of N)
- Proof: (related to Proposition: Limit of Nth Root of a Positive Constant)
- Proof: (related to Proposition: Limit of a Function is Unique If It Exists)
- Proof: (related to Proposition: Limit of a Rational Function)
- Proof: (related to Proposition: Limit of the Constant Function)
- Proof: (related to Proposition: Limit of the Identity Function)
- Proof: (related to Proposition: Limits of General Powers)
- Proof: (related to Proposition: Limits of Logarithm in `$[0,+\infty]$`)
- Proof: (related to Proposition: Limits of Polynomials at Infinity)
- Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral for Step Functions)
- Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral)
- Proof: (related to Proposition: Logarithm to a General Base)
- Proof: (related to Proposition: Minkowski's Inequality for Integral p-norms)
- Proof: (related to Proposition: Minkowski's Inequality)
- Proof: (related to Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable)
- Proof: (related to Proposition: Monotony Criterion)
- Proof: (related to Proposition: Natural Logarithm)
- Proof: (related to Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers)
- Proof: (related to Proposition: Not all Continuous Functions are also Uniformly Continuous)
- Proof: (related to Proposition: Nth Powers)
- Proof: (related to Proposition: Nth Roots of Positive Numbers)
- Proof: (related to Proposition: Oddness of the Sine of a Real Variable)
- Proof: (related to Proposition: Only the Uniform Convergence Preserves Continuity)
- Proof: (related to Proposition: Open Intervals Contain Uncountably Many Irrational Numbers)
- Proof: (related to Proposition: Open Real Intervals are Uncountable)
- Proof: (related to Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable)
- Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain)
- Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions)
- Proof: (related to Proposition: Preservation of Inequalities for Limits of Functions)
- Proof: (related to Proposition: Product of Convegent Complex Sequences)
- Proof: (related to Proposition: Product of Convegent Real Sequences)
- Proof: (related to Proposition: Product of Riemann-integrable Functions is Riemann-integrable)
- Proof: (related to Proposition: Product of a Complex Number and a Convergent Complex Sequence)
- Proof: (related to Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity)
- Proof: (related to Proposition: Product of a Real Number and a Convergent Real Sequence)
- Proof: (related to Proposition: Product of a Real Number and a Convergent Real Series)
- Proof: (related to Proposition: Pythagorean Identity)
- Proof: (related to Proposition: Quotient of Convergent Complex Sequences)
- Proof: (related to Proposition: Quotient of Convergent Real Sequences)
- Proof: (related to Proposition: Raabe's Test)
- Proof: (related to Proposition: Rational Functions are Continuous)
- Proof: (related to Proposition: Rational Numbers are Dense in Real Numbers)
- Proof: (related to Proposition: Rational Powers of Positive Numbers)
- Proof: (related to Proposition: Real Sequences Contain Monotonic Subsequences)
- Proof: (related to Proposition: Rearrangement of Absolutely Convergent Series)
- Proof: (related to Proposition: Rearrangement of Convergent Series)
- Proof: (related to Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence)
- Proof: (related to Proposition: Riemann Integral for Step Functions)
- Proof: (related to Proposition: Riemann Sum Converging To the Riemann Integral)
- Proof: (related to Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions)
- Proof: (related to Proposition: Root Test)
- Proof: (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Proof: (related to Proposition: Square Roots)
- Proof: (related to Proposition: Step Function on Closed Intervals are Riemann-Integrable)
- Proof: (related to Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval)
- Proof: (related to Proposition: Sufficient Condition for a Local Extremum)
- Proof: (related to Proposition: Sum of Arguments of Hyperbolic Cosine)
- Proof: (related to Proposition: Sum of Arguments of Hyperbolic Sine)
- Proof: (related to Proposition: Sum of Convergent Complex Sequences)
- Proof: (related to Proposition: Sum of Convergent Real Sequences)
- Proof: (related to Proposition: Sum of Convergent Real Series)
- Proof: (related to Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty)
- Proof: (related to Proposition: Supremum Norm and Uniform Convergence)
- Proof: (related to Proposition: Taylor's Formula with Remainder Term of Lagrange)
- Proof: (related to Proposition: The distance of complex numbers makes complex numbers a metric space.)
- Proof: (related to Proposition: The distance of real numbers makes real numbers a metric space.)
- Proof: (related to Proposition: Uniform Convergence Criterion of Cauchy)
- Proof: (related to Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series)
- Proof: (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
- Proof: (related to Proposition: Uniqueness Of the Limit of a Sequence)
- Proof: (related to Proposition: Zero of Cosine)
- Proof: (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
- Proof: (related to Proposition: `\(\exp(0)=1\)` (Complex Case))
- Proof: (related to Proposition: `\(\exp(0)=1\)`)
- Proof: (related to Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences)
- Proof: (related to Proposition: n-th Roots of Unity)
- Proof: (related to Theorem: Completeness Principle for Complex Numbers)
- Proof: (related to Theorem: Completeness Principle for Real Numbers)
- Proof: (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
- Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
- Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
- Proof: (related to Theorem: Darboux's Theorem)
- Proof: (related to Theorem: Defining Properties of the Field of Real Numbers)
- Proof: (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
- Proof: (related to Theorem: Fundamental Theorem of Calculus)
- Proof: (related to Theorem: Heine-Borel Theorem)
- Proof: (related to Theorem: Indefinite Integral, Antiderivative)
- Proof: (related to Theorem: Inequality of Weighted Arithmetic Mean)
- Proof: (related to Theorem: Inequality of the Arithmetic Mean)
- Proof: (related to Theorem: Integration by Substitution)
- Proof: (related to Theorem: Intermediate Value Theorem)
- Proof: (related to Theorem: Mean Value Theorem For Riemann Integrals)
- Proof: (related to Theorem: Nested Closed Subset Theorem)
- Proof: (related to Theorem: Partial Integration)
- Proof: (related to Theorem: Reverse Triangle Inequalities)
- Proof: (related to Theorem: Rolle's Theorem)
- Proof: (related to Theorem: Squeezing Theorem for Functions)
- Proof: (related to Theorem: Supremum Property, Infimum Property)
- Proof: (related to Theorem: Triangle Inequality)
- Proof: By Induction (related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
- Proof: By Induction (related to Proposition: Exponential Function of General Base With Natural Exponents)
- Proof: By Induction (related to Proposition: Generalized Bernoulli's Inequality)
- Proof: By Induction (related to Proposition: Generalized Triangle Inequality)
- Proof: By Induction (related to Proposition: Inequality between Powers of `$2$` and Factorials)
- Proof: By Induction (related to Proposition: Inequality between Square Numbers and Powers of `$2$`)
- Proof: By Induction (related to Proposition: Integrals on Adjacent Intervals)
- Proof: By Induction (related to Proposition: Limit of Nth Powers)
- Proof: By Induction (related to Proposition: Limit of a Polynomial)
- Proof: By Induction (related to Proposition: Ratio Test For Absolutely Convergent Complex Series)
- Proof: By Induction (related to Proposition: Ratio Test)
- Proof: By Induction (related to Theorem: Bernoulli's Inequality)
- Proof: By Induction (related to Theorem: De Moivre's Identity, Complex Powers)
- Proof: By Induction (related to Theorem: Every Bounded Real Sequence has a Convergent Subsequence)
- Proof: By Induction (related to Theorem: Inequality Between the Geometric and the Arithmetic Mean)
- Proof: By Induction (related to Theorem: Intermediate Root Value Theorem)
- Proof: By Induction (related to Theorem: Taylor's Formula)

- Combinatorics (35)
- Proof: (related to Corollary: Reciprocity Law of Falling And Rising Factorial Powers)
- Proof: (related to Lemma: Stirling Numbers and Rising Factorial Powers)
- Proof: (related to Proposition: Antidifferences are Unique Up to a Periodic Constant)
- Proof: (related to Proposition: Antidifferences of Some Functions)
- Proof: (related to Proposition: Basic Calculations Involving Indefinite Sums)
- Proof: (related to Proposition: Basic Calculations Involving the Difference Operator)
- Proof: (related to Proposition: Closed Formula For Binomial Coefficients)
- Proof: (related to Proposition: Comparison between the Stirling numbers of the First and Second Kind)
- Proof: (related to Proposition: Difference Operator of Falling Factorial Powers)
- Proof: (related to Proposition: Difference Operator of Powers)
- Proof: (related to Proposition: Factorial Polynomials have a Unique Representation)
- Proof: (related to Proposition: Factorial)
- Proof: (related to Proposition: Factorials and Stirling Numbers of the First Kind)
- Proof: (related to Proposition: Fundamental Counting Principle)
- Proof: (related to Proposition: Indicator Function and Set Operations)
- Proof: (related to Proposition: Inversion Formulas For Stirling Numbers)
- Proof: (related to Proposition: Multinomial Coefficient)
- Proof: (related to Proposition: Number of Ordered n-Tuples in a Set)
- Proof: (related to Proposition: Number of Relations on a Finite Set)
- Proof: (related to Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters)
- Proof: (related to Proposition: Number of Subsets of a Finite Set)
- Proof: (related to Proposition: Recursive Formula for Binomial Coefficients)
- Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the First Kind)
- Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the Second Kind)
- Proof: (related to Proposition: Recursively Defined Arithmetic Functions, Recursion)
- Proof: (related to Proposition: Simple Binomial Identities)
- Proof: (related to Theorem: Approximation of Factorials Using the Stirling Formula)
- Proof: (related to Theorem: Fundamental Theorem of the Difference Calculus)
- Proof: (related to Theorem: Inclusion-Exclusion Principle (Sylvester's Formula))
- Proof: (related to Theorem: Taylor's Formula Using the Difference Operator)
- Proof: By Induction (related to Proposition: Factorial Polynomials vs. Polynomials)
- Proof: By Induction (related to Proposition: Nth Difference Operator)
- Proof: By Induction (related to Proposition: Number of Subsets of a Finite Set)
- Proof: By Induction (related to Theorem: Binomial Theorem)
- Proof: By Induction (related to Theorem: Multinomial Theorem)

- Geometry (532)
- Analytic Geometry (3)
- Euclidean Geometry (529)
- Proof: (related to Proposition: Common Points of Two Distinct Straight Lines in a Plane)
- Proof: (related to Proposition: Common Points of a Plane and a Straight Line Not in the Plane)
- Proof: (related to Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line)
- Proof: (related to Proposition: Plane Determined by two Crossing Straight Lines)
- Elements Euclid (525)
- Book 1 Plane Geometry (68)
- Proof: (Euclid) (related to Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line)
- Proof: (Euclid) (related to Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line)
- Proof: (Euclid) (related to Proposition: 1.18: Angles and Sides in a Triangle I)
- Proof: (Euclid) (related to Proposition: 1.19: Angles and Sides in a Triangle II)
- Proof: (Euclid) (related to Proposition: 1.24: Angles and Sides in a Triangle III)
- Proof: (Euclid) (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Proof: (Euclid) (related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)
- Proof: (Euclid) (related to Proposition: 1.43: Complementary Segments of Parallelograms)
- Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
- Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
- Proof: By Euclid (related to Corollary: Angles of Right Triangle)
- Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
- Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
- Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
- Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
- Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
- Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
- Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
- Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
- Proof: By Euclid (related to Corollary: Rhombus as a Special Case of a Parallelogram)
- Proof: By Euclid (related to Corollary: Similar Triangles)
- Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
- Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
- Proof: By Euclid (related to Corollary: The supplemental angle of a right angle is another right angle.)
- Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
- Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
- Proof: By Euclid (related to Proposition: 1.01: Constructing an Equilateral Triangle)
- Proof: By Euclid (related to Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment)
- Proof: By Euclid (related to Proposition: 1.03: Cutting a Segment at a Given Size)
- Proof: By Euclid (related to Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle)
- Proof: By Euclid (related to Proposition: 1.05: Isosceles Triangles I)
- Proof: By Euclid (related to Proposition: 1.06: Isosceles Triagles II)
- Proof: By Euclid (related to Proposition: 1.07: Uniqueness of Triangles)
- Proof: By Euclid (related to Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles)
- Proof: By Euclid (related to Proposition: 1.09: Bisecting an Angle)
- Proof: By Euclid (related to Proposition: 1.10: Bisecting a Segment)
- Proof: By Euclid (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Proof: By Euclid (related to Proposition: 1.14: Combining Rays to Straight Lines)
- Proof: By Euclid (related to Proposition: 1.15: Opposite Angles on Intersecting Straight Lines)
- Proof: By Euclid (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Proof: By Euclid (related to Proposition: 1.17: The Sum of Two Angles of a Triangle)
- Proof: By Euclid (related to Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality))
- Proof: By Euclid (related to Proposition: 1.21: Triangles within Triangles)
- Proof: By Euclid (related to Proposition: 1.22: Construction of Triangles From Arbitrary Segments)
- Proof: By Euclid (related to Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle)
- Proof: By Euclid (related to Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles)
- Proof: By Euclid (related to Proposition: 1.27: Parallel Lines I)
- Proof: By Euclid (related to Proposition: 1.28: Parallel Lines II)
- Proof: By Euclid (related to Proposition: 1.29: Parallel Lines III)
- Proof: By Euclid (related to Proposition: 1.30: Transitivity of Parallel Lines)
- Proof: By Euclid (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Proof: By Euclid (related to Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram)
- Proof: By Euclid (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Proof: By Euclid (related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
- Proof: By Euclid (related to Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels)
- Proof: By Euclid (related to Proposition: 1.37: Triangles of Equal Area I)
- Proof: By Euclid (related to Proposition: 1.38: Triangles of Equal Area II)
- Proof: By Euclid (related to Proposition: 1.39: Triangles of Equal Area III)
- Proof: By Euclid (related to Proposition: 1.40: Triangles of Equal Area IV)
- Proof: By Euclid (related to Proposition: 1.41: Parallelograms and Triagles)
- Proof: By Euclid (related to Proposition: 1.42: Construction of Parallelograms I)
- Proof: By Euclid (related to Proposition: 1.44: Construction of Parallelograms II)
- Proof: By Euclid (related to Proposition: 1.45: Construction of Parallelograms III)
- Proof: By Euclid (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Proof: By Euclid (related to Proposition: 1.47: Pythagorean Theorem)
- Proof: By Euclid (related to Proposition: 1.48: The Converse of the Pythagorean Theorem)

- Book 2 Geometric Algebra (14)
- Proof: By Euclid (related to Proposition: 2.01: Summing Areas or Rectangles)
- Proof: By Euclid (related to Proposition: 2.02: Square is Sum of Two Rectangles)
- Proof: By Euclid (related to Proposition: 2.03: Rectangle is Sum of Square and Rectangle)
- Proof: By Euclid (related to Proposition: 2.04: Square of Sum)
- Proof: By Euclid (related to Proposition: 2.05: Rectangle is Difference of Two Squares)
- Proof: By Euclid (related to Proposition: 2.06: Square of Sum with One Halved Summand)
- Proof: By Euclid (related to Proposition: 2.07: Sum of Squares)
- Proof: By Euclid (related to Proposition: 2.08: Square of Sum with One Doubled Summand)
- Proof: By Euclid (related to Proposition: 2.09: Sum of Squares of Sum and Difference)
- Proof: By Euclid (related to Proposition: 2.10: Sum of Squares (Half))
- Proof: By Euclid (related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)
- Proof: By Euclid (related to Proposition: 2.12: Law of Cosines (for Obtuse Angles))
- Proof: By Euclid (related to Proposition: 2.13: Law of Cosines (for Acute Angles))
- Proof: By Euclid (related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)

- Book 3 Circles (39)
- Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
- Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Proposition: 3.01: Finding the Center of a given Circle)
- Proof: By Euclid (related to Proposition: 3.02: Chord Lies Inside its Circle)
- Proof: By Euclid (related to Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector)
- Proof: By Euclid (related to Proposition: 3.04: Chords do not Bisect Each Other)
- Proof: By Euclid (related to Proposition: 3.05: Intersecting Circles have Different Centers)
- Proof: By Euclid (related to Proposition: 3.06: Touching Circles have Different Centers)
- Proof: By Euclid (related to Proposition: 3.07: Relative Lengths of Lines Inside Circle)
- Proof: By Euclid (related to Proposition: 3.08: Relative Lengths of Lines Outside Circle)
- Proof: By Euclid (related to Proposition: 3.09: Condition for Point to be Center of Circle)
- Proof: By Euclid (related to Proposition: 3.10: Two Circles have at most Two Points of Intersection)
- Proof: By Euclid (related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
- Proof: By Euclid (related to Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally)
- Proof: By Euclid (related to Proposition: 3.13: Circles Touch at One Point at Most)
- Proof: By Euclid (related to Proposition: 3.14: Equal Chords in Circle)
- Proof: By Euclid (related to Proposition: 3.15: Relative Lengths of Chords of Circles)
- Proof: By Euclid (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Proposition: 3.17: Construction of Tangent from Point to Circle)
- Proof: By Euclid (related to Proposition: 3.18: Radius at Right Angle to Tangent)
- Proof: By Euclid (related to Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center)
- Proof: By Euclid (related to Proposition: 3.20: Inscribed Angle Theorem)
- Proof: By Euclid (related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
- Proof: By Euclid (related to Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral)
- Proof: By Euclid (related to Proposition: 3.23: Segment on Given Base Unique)
- Proof: By Euclid (related to Proposition: 3.24: Similar Segments on Equal Bases are Equal)
- Proof: By Euclid (related to Proposition: 3.25: Construction of Circle from Segment)
- Proof: By Euclid (related to Proposition: 3.26: Equal Angles and Arcs in Equal Circles)
- Proof: By Euclid (related to Proposition: 3.27: Angles on Equal Arcs are Equal)
- Proof: By Euclid (related to Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles)
- Proof: By Euclid (related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
- Proof: By Euclid (related to Proposition: 3.30: Bisection of Arc)
- Proof: By Euclid (related to Proposition: 3.31: Relative Sizes of Angles in Segments)
- Proof: By Euclid (related to Proposition: 3.32: Angles made by Chord with Tangent)
- Proof: By Euclid (related to Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle)
- Proof: By Euclid (related to Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle)
- Proof: By Euclid (related to Proposition: 3.35: Intersecting Chord Theorem)
- Proof: By Euclid (related to Proposition: 3.36: Tangent Secant Theorem)
- Proof: By Euclid (related to Proposition: 3.37: Converse of Tangent Secant Theorem)

- Book 4 Inscription And Circumscription (17)
- Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proof: By Euclid (related to Proposition: 4.01: Fitting Chord Into Circle)
- Proof: By Euclid (related to Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles)
- Proof: By Euclid (related to Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles)
- Proof: By Euclid (related to Proposition: 4.04: Inscribing Circle in Triangle)
- Proof: By Euclid (related to Proposition: 4.05: Circumscribing Circle about Triangle)
- Proof: By Euclid (related to Proposition: 4.06: Inscribing Square in Circle)
- Proof: By Euclid (related to Proposition: 4.07: Circumscribing Square about Circle)
- Proof: By Euclid (related to Proposition: 4.08: Inscribing Circle in Square)
- Proof: By Euclid (related to Proposition: 4.09: Circumscribing Circle about Square)
- Proof: By Euclid (related to Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex)
- Proof: By Euclid (related to Proposition: 4.11: Inscribing Regular Pentagon in Circle)
- Proof: By Euclid (related to Proposition: 4.12: Circumscribing Regular Pentagon about Circle)
- Proof: By Euclid (related to Proposition: 4.13: Inscribing Circle in Regular Pentagon)
- Proof: By Euclid (related to Proposition: 4.14: Circumscribing Circle about Regular Pentagon)
- Proof: By Euclid (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proof: By Euclid (related to Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle)

- Book 5 Proportion (27)
- Proof: By Euclid (related to Corollary: 5.07: Ratios of Equal Magnitudes)
- Proof: By Euclid (related to Corollary: 5.19: Proportional Magnitudes have Proportional Remainders)
- Proof: By Euclid (related to Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition)
- Proof: By Euclid (related to Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition)
- Proof: By Euclid (related to Proposition: 5.03: Multiplication of Numbers is Associative)
- Proof: By Euclid (related to Proposition: 5.04: Multiples of Terms in Equal Ratios)
- Proof: By Euclid (related to Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction)
- Proof: By Euclid (related to Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction)
- Proof: By Euclid (related to Proposition: 5.07: Ratios of Equal Magnitudes)
- Proof: By Euclid (related to Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes)
- Proof: By Euclid (related to Proposition: 5.09: Magnitudes with Same Ratios are Equal)
- Proof: By Euclid (related to Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios)
- Proof: By Euclid (related to Proposition: 5.11: Equality of Ratios is Transitive)
- Proof: By Euclid (related to Proposition: 5.12: Sum of Components of Equal Ratios)
- Proof: By Euclid (related to Proposition: 5.13: Relative Sizes of Proportional Magnitudes)
- Proof: By Euclid (related to Proposition: 5.14: Relative Sizes of Components of Ratios)
- Proof: By Euclid (related to Proposition: 5.15: Ratio Equals its Multiples)
- Proof: By Euclid (related to Proposition: 5.16: Proportional Magnitudes are Proportional Alternately)
- Proof: By Euclid (related to Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated)
- Proof: By Euclid (related to Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded)
- Proof: By Euclid (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
- Proof: By Euclid (related to Proposition: 5.20: Relative Sizes of Successive Ratios)
- Proof: By Euclid (related to Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion)
- Proof: By Euclid (related to Proposition: 5.22: Equality of Ratios Ex Aequali)
- Proof: By Euclid (related to Proposition: 5.23: Equality of Ratios in Perturbed Proportion)
- Proof: By Euclid (related to Proposition: 5.24: Sum of Antecedents of Proportion)
- Proof: By Euclid (related to Proposition: 5.25: Sum of Antecedent and Consequent of Proportion)

- Book 6 Similar Figures (36)
- Proof: By Euclid (related to Corollary: 6.08: Geometric Mean Theorem)
- Proof: By Euclid (related to Corollary: 6.19: Ratio of Areas of Similar Triangles)
- Proof: By Euclid (related to Corollary: 6.20: Similar Polygons are Composed of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base)
- Proof: By Euclid (related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
- Proof: By Euclid (related to Proposition: 6.03: Angle Bisector Theorem)
- Proof: By Euclid (related to Proposition: 6.04: Equiangular Triangles are Similar)
- Proof: By Euclid (related to Proposition: 6.05: Triangles with Proportional Sides are Similar)
- Proof: By Euclid (related to Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar)
- Proof: By Euclid (related to Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar)
- Proof: By Euclid (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.09: Construction of Part of Line)
- Proof: By Euclid (related to Proposition: 6.10: Construction of Similarly Cut Straight Line)
- Proof: By Euclid (related to Proposition: 6.11: Construction of Segment in Squared Ratio)
- Proof: By Euclid (related to Proposition: 6.12: Construction of Fourth Proportional Straight Line)
- Proof: By Euclid (related to Proposition: 6.13: Construction of Mean Proportional)
- Proof: By Euclid (related to Proposition: 6.14: Characterization of Congruent Parallelograms)
- Proof: By Euclid (related to Proposition: 6.15: Characterization of Congruent Triangles)
- Proof: By Euclid (related to Proposition: 6.16: Rectangles Contained by Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.18: Construction of Similar Polygon)
- Proof: By Euclid (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.21: Similarity of Polygons is Transitive)
- Proof: By Euclid (related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms)
- Proof: By Euclid (related to Proposition: 6.24: Parallelograms About Diameter are Similar)
- Proof: By Euclid (related to Proposition: 6.25: Construction of Figure Similar to One and Equal to Another)
- Proof: By Euclid (related to Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter)
- Proof: By Euclid (related to Proposition: 6.27: Similar Parallelogram on Half a Straight Line)
- Proof: By Euclid (related to Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram)
- Proof: By Euclid (related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)
- Proof: By Euclid (related to Proposition: 6.30: Construction of the Inverse Golden Section)
- Proof: By Euclid (related to Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle)
- Proof: By Euclid (related to Proposition: 6.32: Triangles with Two Sides Parallel and Equal)
- Proof: By Euclid (related to Proposition: 6.33: Angles in Circles have Same Ratio as Arcs)

- Book 7 Elementary Number Theory (40)
- Proof: By Euclid (related to Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor)
- Proof: By Euclid (related to Proposition: 7.01: Sufficient Condition for Coprimality)
- Proof: By Euclid (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
- Proof: By Euclid (related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)
- Proof: By Euclid (related to Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers)
- Proof: By Euclid (related to Proposition: 7.05: Divisors Obey Distributive Law (Sum))
- Proof: By Euclid (related to Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum))
- Proof: By Euclid (related to Proposition: 7.07: Divisors Obey Distributive Law (Difference))
- Proof: By Euclid (related to Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference))
- Proof: By Euclid (related to Proposition: 7.09: Alternate Ratios of Equal Fractions)
- Proof: By Euclid (related to Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions)
- Proof: By Euclid (related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
- Proof: By Euclid (related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
- Proof: By Euclid (related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
- Proof: By Euclid (related to Proposition: 7.14: Proportion of Numbers is Transitive)
- Proof: By Euclid (related to Proposition: 7.15: Alternate Ratios of Multiples)
- Proof: By Euclid (related to Proposition: 7.16: Natural Number Multiplication is Commutative)
- Proof: By Euclid (related to Proposition: 7.17: Multiples of Ratios of Numbers)
- Proof: By Euclid (related to Proposition: 7.18: Ratios of Multiples of Numbers)
- Proof: By Euclid (related to Proposition: 7.19: Relation of Ratios to Products)
- Proof: By Euclid (related to Proposition: 7.20: Ratios of Fractions in Lowest Terms)
- Proof: By Euclid (related to Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms)
- Proof: By Euclid (related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime)
- Proof: By Euclid (related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
- Proof: By Euclid (related to Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole)
- Proof: By Euclid (related to Proposition: 7.25: Square of Co-prime Number is Co-prime)
- Proof: By Euclid (related to Proposition: 7.26: Product of Co-prime Pairs is Co-prime)
- Proof: By Euclid (related to Proposition: 7.27: Powers of Co-prime Numbers are Co-prime)
- Proof: By Euclid (related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)
- Proof: By Euclid (related to Proposition: 7.29: Prime not Divisor implies Co-prime)
- Proof: By Euclid (related to Proposition: 7.30: Euclidean Lemma)
- Proof: By Euclid (related to Proposition: 7.31: Existence of Prime Divisors)
- Proof: By Euclid (related to Proposition: 7.32: Natural Number is Prime or has Prime Factor)
- Proof: By Euclid (related to Proposition: 7.33: Least Ratio of Numbers)
- Proof: By Euclid (related to Proposition: 7.34: Existence of Least Common Multiple)
- Proof: By Euclid (related to Proposition: 7.35: Least Common Multiple Divides Common Multiple)
- Proof: By Euclid (related to Proposition: 7.36: Least Common Multiple of Three Numbers)
- Proof: By Euclid (related to Proposition: 7.37: Integer Divided by Divisor is Integer)
- Proof: By Euclid (related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
- Proof: By Euclid (related to Proposition: 7.39: Least Number with Three Given Fractions)

- Book 8 Continued Proportion (28)
- Proof: By Euclid (related to Corollary: 8.02: Construction of Geometric Progression in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes)
- Proof: By Euclid (related to Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios)
- Proof: By Euclid (related to Proposition: 8.05: Ratio of Products of Sides of Plane Numbers)
- Proof: By Euclid (related to Proposition: 8.06: First Element of Geometric Progression not dividing Second)
- Proof: By Euclid (related to Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second)
- Proof: By Euclid (related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
- Proof: By Euclid (related to Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers)
- Proof: By Euclid (related to Proposition: Prop. 8.10: Product of Geometric Progressions from One)
- Proof: By Euclid (related to Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional)
- Proof: By Euclid (related to Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals)
- Proof: By Euclid (related to Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression)
- Proof: By Euclid (related to Proposition: Prop. 8.14: Number divides Number iff Square divides Square)
- Proof: By Euclid (related to Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
- Proof: By Euclid (related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional)
- Proof: By Euclid (related to Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals)
- Proof: By Euclid (related to Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane)
- Proof: By Euclid (related to Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid)
- Proof: By Euclid (related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)
- Proof: By Euclid (related to Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)
- Proof: By Euclid (related to Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares)
- Proof: By Euclid (related to Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes)

- Book 9 Number Theory Applications (37)
- Proof: By Euclid (related to Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Proof: By Euclid (related to Proposition: 9.35: Sum of Geometric Progression)
- Proof: By Euclid (related to Proposition: 9.36: Theorem of Even Perfect Numbers (First Part))
- Proof: By Euclid (related to Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square)
- Proof: By Euclid (related to Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.03: Square of Cube Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.06: Number Squared making Cube is itself Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number)
- Proof: By Euclid (related to Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Proof: By Euclid (related to Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime)
- Proof: By Euclid (related to Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime)
- Proof: By Euclid (related to Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element)
- Proof: By Euclid (related to Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional)
- Proof: By Euclid (related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second)
- Proof: By Euclid (related to Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.20: Infinite Number of Primes)
- Proof: By Euclid (related to Proposition: Prop. 9.21: Sum of Even Numbers is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.24: Even Number minus Even Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.26: Odd Number minus Odd Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.27: Odd Number minus Even Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)
- Proof: By Euclid (related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)
- Proof: By Euclid (related to Proposition: Prop. 9.32: Power of Two is Even-Times Even Only)
- Proof: By Euclid (related to Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd)
- Proof: By Euclid (related to Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic)

- Book 10 Incommensurable Magnitudes (131)
- Proof: 547 BC (related to Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Proof: 569 BC (related to Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Proof: 600 BC (related to Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Proof: By Euclid (related to Corollary: Cor. 10.009: Commensurability of Squares)
- Proof: By Euclid (related to Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial)
- Proof: By Euclid (related to Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order)
- Proof: By Euclid (related to Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas)
- Proof: By Euclid (related to Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
- Proof: By Euclid (related to Lemma: Lem. 10.021: Medial is Irrational)
- Proof: By Euclid (related to Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square)
- Proof: By Euclid (related to Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square)
- Proof: By Euclid (related to Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II)
- Proof: By Euclid (related to Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational)
- Proof: By Euclid (related to Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas)
- Proof: By Euclid (related to Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them)
- Proof: By Euclid (related to Lemma: Lem. 10.13: Finding Pythagorean Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number)
- Proof: By Euclid (related to Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm)
- Proof: By Euclid (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio)
- Proof: By Euclid (related to Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.009: Commensurability of Squares)
- Proof: By Euclid (related to Proposition: Prop. 10.010: Construction of Incommensurable Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.012: Commensurability is Transitive Relation)
- Proof: By Euclid (related to Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude)
- Proof: By Euclid (related to Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation)
- Proof: By Euclid (related to Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation)
- Proof: By Euclid (related to Proposition: Prop. 10.019: Product of Rational Numbers is Rational)
- Proof: By Euclid (related to Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational)
- Proof: By Euclid (related to Proposition: Prop. 10.021: Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.022: Square on Medial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
- Proof: By Euclid (related to Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial)
- Proof: By Euclid (related to Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square)
- Proof: By Euclid (related to Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area)
- Proof: By Euclid (related to Proposition: Prop. 10.027: Construction of Components of First Bimedial)
- Proof: By Euclid (related to Proposition: Prop. 10.028: Construction of Components of Second Bimedial)
- Proof: By Euclid (related to Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I)
- Proof: By Euclid (related to Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II)
- Proof: By Euclid (related to Proposition: Prop. 10.033: Construction of Components of Major)
- Proof: By Euclid (related to Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area)
- Proof:

- Book 1 Plane Geometry (68)

- Algebra (73)