| Proof of the Chinese remainder theorem for non-coprime moduli | Proof | Number theory |
| Proof of the existence and uniqueness of a fraction in lowest terms | Proof | Number theory |
| Proof of the orbit-stabiliser theorem | Proof | Group theory and generalizations |
| Proof of the continuity of the sum of a uniformly convergent series of functions | Proof | Sequences, series, summability |
| Proof of the existence of the infimum by negation | Proof | Real functions |
| Proof of the divisibility rule for 11 | Proof | Number theory |
| Proof of the existence of a complementary subspace by extending a basis | Proof | Linear and multilinear algebra; matrix theory |
| Proof that a tangent is perpendicular to the radius | Proof | Geometry |
| Proof of the second supplement to quadratic reciprocity from Gauss's lemma | Proof | Number theory |
| Proof of the volume of a ball in R^n by slicing | Proof | Measure and integration |
| Proof of the edge bound for planar graphs | Proof | Combinatorics |
| Proof that a countable intersection of dense open sets in a Baire space is dense | Proof | General topology |
| Proof of the variation of constants formula for linear systems | Proof | Ordinary differential equations |
| Proof of the divergence part of the ratio test | Proof | Sequences, series, summability |
| Proof that the order of a finite group is the index times the order of a subgroup | Proof | Group theory and generalizations |
| Proof of stars and bars by a bijection with subsets | Proof | Combinatorics |
| Proof of the associativity of addition by induction | Proof | Mathematical logic and foundations |
| Proof of the associativity of composition | Proof | Mathematical logic and foundations |
| Proof that if 2^n + 1 is prime then n is a power of 2 | Proof | Number theory |
| Proof of the equivalence of norms in finite dimension by compactness of the unit sphere | Proof | Functional analysis |
| Proof of the description of the closure in a metric space | Proof | General topology |
| Proof that path-connected spaces are connected | Proof | General topology |
| Proof that trees are bipartite | Proof | Combinatorics |
| Proof of the determinant of the inverse | Proof | Linear and multilinear algebra; matrix theory |
| Proof that every spanning set contains a basis, by Zorn's lemma | Proof | Linear and multilinear algebra; matrix theory |
| Proof that a polynomial over a field has at most as many roots as its degree | Proof | Field theory and polynomials |
| Proof of the remainder theorem | Proof | Commutative algebra |
| Proof of the formula for the totient of a prime | Proof | Number theory |
| Proof that the determinant of the identity matrix is one | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the continuity of the inverse of a strictly monotone continuous function | Proof | Real functions |
| Proof of the estimation lemma | Proof | Functions of a complex variable |
| Proof that a cyclic group of order n has phi(d) elements of order d | Proof | Group theory and generalizations |
| Proof of Fermat's theorem on stationary points | Proof | Real functions |
| Proof that limits preserve non-strict inequalities | Proof | Sequences, series, summability |
| Proof that a linear map is injective iff its kernel is trivial | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the inverse function theorem by the contraction mapping principle | Proof | Real functions |
| Proof of the polar decomposition from the singular value decomposition | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the formula for the distance from a point to a plane | Proof | Geometry |
| Proof that extensions of solvable groups are solvable | Proof | Group theory and generalizations |
| Proof that metrizable spaces are normal by distance functions | Proof | General topology |
| Proof that every linearly independent set extends to a basis, by Zorn's lemma | Proof | Linear and multilinear algebra; matrix theory |
| Proof that connected subsets of the real line are intervals | Proof | General topology |
| Proof that closed sets contain their limits | Proof | General topology |
| Proof of the freshman's dream | Proof | Commutative algebra |
| Proof of Cassini's identity by induction | Proof | Number theory |
| Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial | Proof | Field theory and polynomials |
| Proof of De Morgan's laws in a Boolean algebra | Proof | Order, lattices, ordered algebraic structures |
| Proof that the number of Sylow subgroups divides the index | Proof | Group theory and generalizations |
| Proof of simultaneous diagonalisation of commuting diagonalisable matrices | Proof | Linear and multilinear algebra; matrix theory |
| Proof that 1 + x is at most exp x | Proof | Special functions |
| Proof of the linearity of the integral | Proof | Real functions |
| Proof of the formula for the characteristic function of the normal distribution | Proof | Probability theory and stochastic processes |
| Proof of the factorial formula for binomial coefficients by induction | Proof | Combinatorics |
| Proof that congruence is compatible with multiplication | Proof | Number theory |
| Proof that a finite group is nilpotent iff it is the product of its Sylow subgroups | Proof | Group theory and generalizations |
| Proof of the dimension of the dual space by the dual basis | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the isosceles triangle theorem with the dot product | Proof | Geometry |
| Proof that a function with positive derivative is strictly increasing | Proof | Real functions |
| Proof of integration by substitution | Proof | Real functions |
| Proof of the second Borel–Cantelli lemma | Proof | Probability theory and stochastic processes |
| Proof of Binet's formula by induction | Proof | Number theory |
| Proof that independent random variables are uncorrelated | Proof | Probability theory and stochastic processes |
| Proof that sin x / x tends to 1 as a derivative at 0 | Proof | Special functions |
| Proof that the modulus of the conjugate equals the modulus | Proof | Functions of a complex variable |
| Proof of the existence of splitting fields | Proof | Field theory and polynomials |
| Proof of the recurrence for Bell numbers by the block of the last element | Proof | Number theory |
| Proof of the determinant of a scalar multiple | Proof | Linear and multilinear algebra; matrix theory |
| Proof that the order modulo p divides p − 1 | Proof | Number theory |
| Proof of Hall's marriage theorem by induction on the number of sets | Proof | Combinatorics |
| Proof of the formula for coordinates in an orthonormal basis | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the hockey-stick identity by induction | Proof | Combinatorics |
| Proof of Zorn's lemma | Proof | Mathematical logic and foundations |
| Proof of the third isomorphism theorem for rings | Proof | Associative rings and algebras |
| Proof that metric spaces are Hausdorff | Proof | General topology |
| Proof of Lebesgue's criterion for Riemann integrability | Proof | Real functions |
| Proof of Friedberg–Muchnik theorem below C | Proof | |
| Proof of the matrix determinant lemma from Sylvester's determinant theorem | Proof | Linear and multilinear algebra; matrix theory |
| Proof that the matrix of the dual map is the transpose | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the formula for the determinant of a 2 by 2 matrix | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the product of exponentials by the Cauchy product | Proof | Special functions |
| Proof of the comparison of a series and an integral | Proof | Real functions |
| Proof of Tutte's theorem via an edge-maximal counterexample | Proof | Combinatorics |
| Proof of Ptolemy's theorem by complex numbers | Proof | Geometry |
| Proof that the order of a finite field is a prime power | Proof | Field theory and polynomials |
| Proof of the Cholesky decomposition by induction on the size | Proof | Linear and multilinear algebra; matrix theory |
| Proof of unique paths in trees | Proof | Combinatorics |
| Proof that the units of the Gaussian integers are 1, -1, i and -i | Proof | Number theory |
| Proof that an iterate of the Picard operator is a contraction | Proof | Ordinary differential equations |
| Proof that subgroups of abelian groups are normal | Proof | Group theory and generalizations |
| Proof of the quadratic formula by completing the square | Proof | Field theory and polynomials |
| Proof that the derivative gives the directional derivatives | Proof | Real functions |
| Proof that the rationals are countable | Proof | Mathematical logic and foundations |
| Proof of the second fundamental theorem of calculus | Proof | Real functions |
| Proof of global existence for globally Lipschitz vector fields by steps of uniform length | Proof | Ordinary differential equations |
| Proof of the spectral theorem for normal matrices by deflation | Proof | Linear and multilinear algebra; matrix theory |
| Proof that the complex numbers form a field | Proof | Field theory and polynomials |
| Proof of the intermediate value theorem by the supremum | Proof | Real functions |
| Proof of the binomial theorem by induction | Proof | Combinatorics |
| Proof of the formula for the Laplace transform of a derivative by integration by parts | Proof | Integral transforms, operational calculus |
| Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms | Proof | Real functions |
| Proof of Taylor's theorem with the Lagrange remainder by the Cauchy mean value theorem | Proof | Approximations and expansions |
| Proof that differentiable functions are continuous | Proof | Real functions |
| Proof that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism | Proof | General topology |
| Proof of the Poisson limit theorem | Proof | Probability theory and stochastic processes |
| Proof of the Stolz–Cesàro theorem | Proof | Sequences, series, summability |
| Proof of the decomposition of a permutation into disjoint cycles | Proof | Group theory and generalizations |
| Proof that a linear map is determined by its values on a basis | Proof | Linear and multilinear algebra; matrix theory |
| Proof of Liouville's theorem on Diophantine approximation | Proof | Number theory |
| Proof of the reduction of a higher-order equation to a first-order system | Proof | Ordinary differential equations |
| Proof that closed subsets of compact sets are compact | Proof | General topology |
| Proof of the integral test | Proof | Sequences, series, summability |
| Proof of Heron's formula from the law of cosines | Proof | Geometry |
| Proof of Abel's test from Dirichlet's test | Proof | Sequences, series, summability |
| Proof of cosh^2 - sinh^2 = 1 | Proof | Special functions |
| Proof of the spectral theorem for Hermitian matrices | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the dimension formula for a direct product | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the diagonalisation of a quadratic form by an orthogonal basis | Proof | Number theory |
| Proof that the minimal polynomial divides the characteristic polynomial | Proof | Linear and multilinear algebra; matrix theory |
| Proof that quotients of solvable groups are solvable | Proof | Group theory and generalizations |
| Proof that one-sided limits of a monotone function exist | Proof | Real functions |
| Proof that a finite abelian group is cyclic iff its exponent equals its order | Proof | Group theory and generalizations |
| Proof of triangularisation by induction on the size | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the addition rule for two events | Proof | Probability theory and stochastic processes |
| Proof that angles in the same segment are equal | Proof | Geometry |
| Proof of Dini's theorem by compactness | Proof | Sequences, series, summability |
| Proof of term-by-term integration | Proof | Sequences, series, summability |
| Proof of the existence of finite fields of every prime power order | Proof | Field theory and polynomials |
| Proof of the law of total probability | Proof | Probability theory and stochastic processes |
| Proof of Sylvester's determinant theorem by two block factorizations | Proof | Linear and multilinear algebra; matrix theory |
| Proof of Hensel's lemma by Newton steps | Proof | Number theory |
| Proof of Cayley's theorem by left multiplication | Proof | Group theory and generalizations |
| Proof that nilpotent groups are solvable | Proof | Group theory and generalizations |
| Proof of the product of a matrix and its adjugate by Laplace expansion | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the formula for the order of a power | Proof | Group theory and generalizations |
| Proof of Dirichlet's approximation theorem by the pigeonhole principle | Proof | Number theory |
| Proof of Taylor's theorem in several variables by restriction to a line | Proof | Real functions |
| Proof of the power series of arctangent by integrating a finite geometric sum | Proof | Functions of a complex variable |
| Proof of the Rouché–Capelli theorem | Proof | Linear and multilinear algebra; matrix theory |
| Proof that the order of a cyclic subgroup equals the order of its generator | Proof | Group theory and generalizations |
| Proof of the existence of primitive roots modulo a prime by counting elements of each order | Proof | Number theory |
| Proof of the volume of a ball in R^3 from the n-dimensional formula | Proof | Real functions |
| Proof of the rule of sum | Proof | Combinatorics |
| Proof that abelian groups are solvable | Proof | Group theory and generalizations |
| Proof of the determinant of a triangular matrix by the Leibniz formula | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the impossibility of trisecting the angle | Proof | Field theory and polynomials |
| Proof that harmonic numbers grow like log n | Proof | Number theory |
| Proof that convergence in probability implies convergence in distribution | Proof | Probability theory and stochastic processes |
| Proof of the sequence characterisation of uniform continuity | Proof | General topology |
| Proof that sine and cosine are 2π-periodic with smallest period 2π | Proof | Special functions |
| Proof of the epsilon-delta form of the limit of a function | Proof | Real functions |
Math 2.0