| Proof of the law of total variance | Proof | Probability theory and stochastic processes |
| Proof of Ore's theorem | Proof | Combinatorics |
| Proof of the dimension formula for a tensor product | Proof | Linear and multilinear algebra; matrix theory |
| Proof of Pépin's test | Proof | Number theory |
| Proof of the Neyman–Pearson lemma | Proof | Statistics |
| Proof of the fundamental theorem of algebra by Liouville's theorem | Proof | Field theory and polynomials |
| Proof of the Laplacian as a sum of second partial derivatives | Proof | Partial differential equations |
| Proof of Fubini's theorem for Riemann integrals on a rectangle | Proof | Real functions |
| Proof of the Wronskian test for linear independence of solutions | Proof | Ordinary differential equations |
| Proof of the fundamental theorem of symmetric polynomials by leading monomials | Proof | Combinatorics |
| Proof that Z/nZ is a field if and only if n is prime | Proof | Number theory |
| Proof that equivalence classes form a partition | Proof | Mathematical logic and foundations |
| Proof that the solution space of a homogeneous linear equation has dimension n | Proof | Ordinary differential equations |
| Proof of the power rule by induction | Proof | Real functions |
| Proof of the number of k-element subsets by induction | Proof | Combinatorics |
| Proof that absolutely convergent series converge | Proof | Sequences, series, summability |
| Proof of the triangle inequality for norms | Proof | Functional analysis |
| Proof that g to the n equals 1 iff the order divides n | Proof | Group theory and generalizations |
| Proof of Euler's identity | Proof | Special functions |
| Proof of the sum of a finite geometric series | Proof | Sequences, series, summability |
| Proof of variance under scaling | Proof | Probability theory and stochastic processes |
| Proof of De Morgan's laws for sets | Proof | Mathematical logic and foundations |
| Proof of the description of the nth roots of unity | Proof | Number theory |
| Proof that convergent sequences are Cauchy | Proof | Sequences, series, summability |
| Proof that every infinite set has a countably infinite subset | Proof | Mathematical logic and foundations |
| Proof that a complex number times its conjugate is the square of its modulus | Proof | Functions of a complex variable |
| Proof of the formula for the number of derangements by inclusion–exclusion | Proof | Combinatorics |
| Proof of De Morgan's laws by truth tables | Proof | Mathematical logic and foundations |
| Proof of the classification of Pythagorean triples | Proof | Number theory |
| Proof of the dimension formula for the space of linear maps | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the Lagrange multiplier theorem by the implicit function theorem | Proof | Operations research, mathematical programming |
| Proof of the parallelogram law | Proof | Functional analysis |
| Proof of the existence of an orthonormal basis by the Gram–Schmidt process | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the intermediate value theorem for connected spaces | Proof | General topology |
| Proof that the Gaussian integers are a Euclidean domain | Proof | Number theory |
| Proof that time translates of solutions of an autonomous system are solutions | Proof | Ordinary differential equations |
| Proof of the dimension formula for a quotient space | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the correspondence between equivalence relations and partitions | Proof | Mathematical logic and foundations |
| Proof that a nontrivial finite p-group has nontrivial centre by the class equation | Proof | Group theory and generalizations |
| Proof of the power series of sine | Proof | Special functions |
| Proof of the second Sylow theorem | Proof | Group theory and generalizations |
| Proof that G is abelian if G modulo its centre is cyclic | Proof | Group theory and generalizations |
| Proof of the tangent–secant theorem by the power of a point | Proof | Geometry |
| Proof of the Cauchy–Schwarz inequality | Proof | Linear and multilinear algebra; matrix theory |
| Proof that a matrix is diagonalisable iff it has a basis of eigenvectors | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the description of the automorphism group of a cyclic group | Proof | Group theory and generalizations |
| Proof that every principal ideal domain is a unique factorisation domain | Proof | Commutative algebra |
| Proof that the orthogonal projection is the closest point | Proof | Functional analysis |
| Proof of the Heine–Borel theorem by sequential compactness | Proof | General topology |
| Proof of Mertens' theorem on Cauchy products | Proof | Sequences, series, summability |
| Proof that there is a natural number n with 1/n below any positive epsilon | Proof | Real functions |
| Proof of the chain rule in several variables | Proof | Real functions |
| Proof that a composition of injections is injective | Proof | Mathematical logic and foundations |
| Proof of the chain rule with an auxiliary difference quotient | Proof | Real functions |
| Proof of the irrationality of e, after Fourier | Proof | Number theory |
| Proof of the change of basis formula | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the third isomorphism theorem for vector spaces | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the uniqueness of splitting fields | Proof | Field theory and polynomials |
| Proof of the integral of the exponential function | Proof | Real functions |
| Proof of Grassmann's formula by extending a basis of the intersection | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the distributivity of matrix multiplication | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the Pythagorean theorem in an inner product space | Proof | Functional analysis |
| Proof of Fermat's theorem on sums of two squares by Thue's lemma | Proof | Number theory |
| Proof that the continued fraction of a real number is finite exactly when the number is rational | Proof | Number theory |
| Proof that a bounded derivative on a convex set implies Lipschitz | Proof | Real functions |
| Proof of the irrationality of the square root of 2 by lowest terms | Proof | Number theory |
| Proof of the limit of the cumulative distribution function at infinity | Proof | Probability theory and stochastic processes |
| Proof of induction on the integers | Proof | Number theory |
| Proof of side-side-side congruence with orthonormal frames | Proof | Geometry |
| Proof of the Riesz representation theorem in finite dimensions | Proof | Functional analysis |
| Proof of the gradient theorem | Proof | Real functions |
| Proof that the eigenvalues of a self-adjoint operator are real | Proof | Operator theory |
| Proof that the constructible numbers form the smallest subfield of the complex numbers closed under square roots | Proof | Field theory and polynomials |
| Proof of the divergence theorem on a rectangle in the plane | Proof | Real functions |
| Proof of the probability of the complement | Proof | Probability theory and stochastic processes |
| Proof that the constant coefficient of the characteristic polynomial is a signed determinant | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the Frobenius coin problem for two coins | Proof | Number theory |
| Proof that a union of open sets of a metric space is open | Proof | General topology |
| Proof of Boole's inequality by disjoint pieces | Proof | Probability theory and stochastic processes |
| Proof that geometric solutions come from roots of the characteristic polynomial | Proof | Difference and functional equations |
| Proof of the Cauchy criterion for series | Proof | Sequences, series, summability |
| Proof of variance under translation | Proof | Probability theory and stochastic processes |
| Proof of Fermat's right triangle theorem by infinite descent | Proof | Number theory |
| Proof of the tower law | Proof | Field theory and polynomials |
| Proof that exp(2 pi i/n) is a primitive root of unity | Proof | Number theory |
| Proof that the logarithm is strictly concave | Proof | Real functions |
| Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem | Proof | Sequences, series, summability |
| Proof of the multiplicativity of the Möbius function | Proof | Number theory |
| Proof that the matrix of a composition is the product of the matrices | Proof | Linear and multilinear algebra; matrix theory |
| Proof that a monotone function has countably many discontinuities | Proof | Real functions |
| Proof of Euclid's lemma | Proof | Number theory |
| Proof that the coordinate space F^n has dimension n | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the reverse triangle inequality | Proof | Real functions |
| Proof that the determinant is multiplicative by expanding the rows | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the uniqueness of the algebraic closure | Proof | Field theory and polynomials |
| Proof that every polynomial has a root in some extension field | Proof | Field theory and polynomials |
| Proof that an intersection of closed sets of a metric space is closed | Proof | General topology |
| Proof that a countable union of countable sets is countable | Proof | Mathematical logic and foundations |
| Proof that permuting rows multiplies the determinant by the sign | Proof | Linear and multilinear algebra; matrix theory |
| Proof that the exponential series has infinite radius of convergence | Proof | Functions of a complex variable |
| Proof of the limit of a quotient of sequences | Proof | Sequences, series, summability |
| Proof of the inequality of arithmetic and geometric means by induction | Proof | Real functions |
| Proof that closed subsets of complete spaces are complete | Proof | General topology |
| Proof of the addition formula for Fibonacci numbers by induction | Proof | Number theory |
| Proof of the derivative as a little-o approximation | Proof | Real functions |
| Proof of the second-order Taylor expansion | Proof | Real functions |
| Proof that the gradient is orthogonal to level sets | Proof | Real functions |
| Proof that cyclic groups are abelian | Proof | Group theory and generalizations |
| Proof that disjoint cycles commute | Proof | Group theory and generalizations |
| Proof that R[x] is an integral domain iff R is | Proof | Commutative algebra |
| Proof of Urysohn's lemma by dyadic open sets | Proof | General topology |
| Proof that cos pi = -1 and sin pi = 0 | Proof | Special functions |
| Proof that g to the order of the group is the identity | Proof | Group theory and generalizations |
| Proof that a matrix with a zero row has determinant zero | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the Brahmagupta–Fibonacci identity | Proof | Number theory |
| Erdős's proof of the divergence of the sum of the reciprocals of the primes | Proof | Number theory |
| Proof that the derivative vanishes at a local extremum | Proof | Real functions |
| Proof of Stone's representation theorem for finite Boolean algebras | Proof | Order, lattices, ordered algebraic structures |
| Proof of the equality of complex numbers | Proof | Functions of a complex variable |
| Proof that conjugacy classes of the symmetric group correspond to cycle types | Proof | Group theory and generalizations |
| Proof of Abel's theorem by summation by parts | Proof | Sequences, series, summability |
| Proof that the inclusion of a subspace is continuous | Proof | General topology |
| Proof that continuous functions on a closed bounded interval are uniformly continuous | Proof | Real functions |
| Proof of Cayley's formula by Prüfer sequences | Proof | Combinatorics |
| Proof that scaling a row scales the determinant | Proof | Linear and multilinear algebra; matrix theory |
| Proof that a matrix is invertible iff its determinant is a unit | Proof | Linear and multilinear algebra; matrix theory |
| Proof that a linear map between normed spaces is continuous if and only if it is bounded | Proof | Operator theory |
| Proof that the trace is the sum of the eigenvalues by comparing coefficients | Proof | Linear and multilinear algebra; matrix theory |
| Proof of the determinant of a block triangular matrix by the Leibniz formula | Proof | Linear and multilinear algebra; matrix theory |
| Proof that K5 and K3,3 are not planar | Proof | Combinatorics |
| Proof that a function with nonnegative derivative is increasing | Proof | Real functions |
| Proof that the inner automorphism group is isomorphic to G modulo its centre | Proof | Group theory and generalizations |
| Proof that a function is bijective iff it has a two-sided inverse | Proof | Mathematical logic and foundations |
| Proof of Kummer's theorem from Legendre's formula | Proof | Number theory |
| Proof of Parseval's identity | Proof | Functional analysis |
| Proof of the third isomorphism theorem | Proof | Group theory and generalizations |
| Proof that the complex numbers cannot be ordered | Proof | Field theory and polynomials |
| Proof of transfinite induction from well-foundedness | Proof | Mathematical logic and foundations |
| Proof that finite extensions are algebraic | Proof | Field theory and polynomials |
| Proof of the Lucas–Lehmer test | Proof | Number theory |
| Proof that the reals form an ordered field, with the Cauchy sequence construction | Proof | Field theory and polynomials |
| Proof that the Galois group of a finite Galois extension has order equal to the degree | Proof | Field theory and polynomials |
| Proof of the root test | Proof | Sequences, series, summability |
| Proof of continuity via closed sets | Proof | General topology |
| Proof of the order of the symmetric group | Proof | Group theory and generalizations |
| Proof that a graph is bipartite iff it has no odd cycle | Proof | Combinatorics |
| Proof that the degree of a simple extension is the degree of the minimal polynomial | Proof | Field theory and polynomials |
| Proof that reversing a path changes the sign of a line integral | Proof | Real functions |
| Proof of the order of the alternating group | Proof | Group theory and generalizations |
| Proof that complete subspaces are closed | Proof | General topology |
Math 2.0