| Generalised Stokes theorem | Theorem | Global analysis, analysis on manifolds |
| Green's theorem | Theorem | Real functions |
| Solutions for repeated roots | Theorem | Ordinary differential equations |
| Axiom of choice | Axiom | Mathematical logic and foundations |
| Rouché–Capelli theorem | Theorem | Linear and multilinear algebra; matrix theory |
| Limit comparison test | Theorem | Sequences, series, summability |
| Totient of a prime | Proposition | Number theory |
| Eigenvalues of a positive definite matrix are positive | Theorem | Linear and multilinear algebra; matrix theory |
| Injective iff it has a left inverse | Proposition | Mathematical logic and foundations |
| Modulus is multiplicative | Proposition | Functions of a complex variable |
| Mean value property | Theorem | Potential theory |
| Solutions of Pell's equation are powers of the fundamental solution | Theorem | Number theory |
| Polar decomposition | Theorem | Linear and multilinear algebra; matrix theory |
| Conjugacy classes of the symmetric group correspond to cycle types | Theorem | Group theory and generalizations |
| The symmetric group on five or more letters is not solvable | Theorem | Group theory and generalizations |
| Congruence is compatible with powers | Corollary | Number theory |
| Intersecting secants theorem | Theorem | Geometry |
| Lyapunov's stability theorem | Theorem | Ordinary differential equations |
| Associativity of composition | Proposition | Mathematical logic and foundations |
| Kuratowski's theorem | Theorem | Combinatorics |
| Rearrangements of an absolutely convergent series have the same sum | Theorem | Sequences, series, summability |
| Logarithm of a quotient | Corollary | Special functions |
| Sperner's theorem | Theorem | Combinatorics |
| Taylor's theorem in several variables | Theorem | Real functions |
| Determinant of a block triangular matrix | Proposition | Linear and multilinear algebra; matrix theory |
| Ratio formula for the radius of convergence | Proposition | Functions of a complex variable |
| Vector calculus identities | Theorem | Real functions |
| Sign is a homomorphism | Theorem | Group theory and generalizations |
| The cumulative distribution function determines the distribution | Theorem | Probability theory and stochastic processes |
| Turán's theorem | Theorem | Combinatorics |
| Eigenspaces of a self-adjoint operator are orthogonal | Proposition | Operator theory |
| Cauchy's estimates | Corollary | Functions of a complex variable |
| Abelian simple groups have prime order | Proposition | Group theory and generalizations |
| Cauchy's theorem | Theorem | Group theory and generalizations |
| Lindemann–Weierstrass theorem | Theorem | Number theory |
| Convolution theorem for the Laplace transform | Theorem | Integral transforms, operational calculus |
| Fundamental theorem of algebra | Theorem | Field theory and polynomials |
| Kőnig's theorem | Theorem | Combinatorics |
| Subgroups of abelian groups are normal | Proposition | Group theory and generalizations |
| Order of the general linear group over a finite field | Theorem | Group theory and generalizations |
| Weierstrass approximation theorem | Theorem | Approximations and expansions |
| Estimate for approximate solutions | Theorem | Ordinary differential equations |
| Tychonoff's theorem | Theorem | General topology |
| An orthonormal family is linearly independent | Proposition | Linear and multilinear algebra; matrix theory |
| L'Hôpital's rule | Theorem | Real functions |
| Divergence theorem | Theorem | Real functions |
| Legendre's theorem on continued fractions | Theorem | Number theory |
| Lifting-the-exponent lemma | Lemma | Number theory |
| Power rule | Proposition | Real functions |
| Laplace transform of a derivative | Proposition | Integral transforms, operational calculus |
| Distance from a point to a plane | Proposition | Geometry |
| Lipschitz functions are uniformly continuous | Proposition | Real functions |
| Determinant of a triangular matrix | Proposition | Linear and multilinear algebra; matrix theory |
| Integral of 1/x | Proposition | Real functions |
| Brahmagupta–Fibonacci identity | Proposition | Number theory |
| Logarithm of a product | Proposition | Special functions |
| Minkowski's theorem | Theorem | Number theory |
| Factor theorem | Theorem | Commutative algebra |
| Determinant of a matrix with two equal rows | Proposition | Linear and multilinear algebra; matrix theory |
| Complement of an open set is closed | Proposition | General topology |
| Isolated zeros theorem | Theorem | Functions of a complex variable |
| Compactness and sequential compactness agree in metric spaces | Theorem | General topology |
| Continuous image of a compact set is compact | Proposition | General topology |
| Inverse function theorem | Theorem | Real functions |
| General Leibniz rule | Theorem | Real functions |
| Determinant via the Schur complement | Theorem | Linear and multilinear algebra; matrix theory |
| Sum of binomial coefficients | Theorem | Combinatorics |
| Subgroups of cyclic groups are cyclic | Theorem | Group theory and generalizations |
| Distributive law | Theorem | Mathematical logic and foundations |
| Stirling's formula | Theorem | Approximations and expansions |
| Bolzano–Weierstrass theorem | Theorem | Sequences, series, summability |
| Classification of Pythagorean triples | Theorem | Number theory |
| The alternating group is generated by three-cycles | Proposition | Group theory and generalizations |
| A polynomial over a field has at most as many roots as its degree | Theorem | Field theory and polynomials |
| One-sided limits of a monotone function exist | Theorem | Real functions |
| Hausdorff maximal principle | Theorem | Mathematical logic and foundations |
| Determinant of a linear map is the determinant of its matrix | Proposition | Linear and multilinear algebra; matrix theory |
| Strong induction | Theorem | Mathematical logic and foundations |
| The exponential function is strictly increasing | Proposition | Special functions |
| Ceva's theorem | Theorem | Geometry |
| Variance under scaling | Proposition | Probability theory and stochastic processes |
| Reduction modulo p test for irreducibility | Theorem | Field theory and polynomials |
| Radius of convergence of the binomial series | Proposition | Functions of a complex variable |
| Continuous partial derivatives imply differentiability | Theorem | Real functions |
| Derivative of x^a | Theorem | Real functions |
| Lagrange's four-square theorem | Theorem | Number theory |
| Quadratic formula | Theorem | Field theory and polynomials |
| Derivative of an inverse function | Theorem | Real functions |
| Linearity of differentiation | Theorem | Real functions |
| Intercept theorem | Theorem | Geometry |
| Bernoulli's inequality | Theorem | Real functions |
| Fermat's little theorem | Theorem | Number theory |
| Curl of a gradient is zero | Proposition | Real functions |
| Cayley's formula | Theorem | Combinatorics |
| The number of Sylow subgroups divides the index | Theorem | Group theory and generalizations |
| Four colour theorem | Theorem | Combinatorics |
| Every linearly independent set extends to a basis | Theorem | Linear and multilinear algebra; matrix theory |
| Classification of the Platonic solids | Theorem | Geometry |
| Existence of inverses modulo n | Theorem | Number theory |
| The only ideals of a field are zero and the whole field | Proposition | Field theory and polynomials |
| Universal property of the product topology | Theorem | General topology |
| Euler's criterion | Theorem | Number theory |
| Vieta's formulas | Theorem | Field theory and polynomials |
| Approximation property of the supremum | Proposition | Real functions |
| Schwarz lemma | Lemma | Functions of a complex variable |
| Monotonicity of the integral | Theorem | Real functions |
| A finite abelian group is cyclic iff its exponent equals its order | Proposition | Group theory and generalizations |
| Orthogonal projection is the closest point | Theorem | Functional analysis |
| Epsilon characterisation of Cauchy sequences | Proposition | General topology |
| Division algorithm for polynomials | Theorem | Commutative algebra |
| Derivative of a polynomial | Corollary | Real functions |
| Wronskian test for linear independence of solutions | Theorem | Ordinary differential equations |
| Every sequence has a monotone subsequence | Lemma | Sequences, series, summability |
| A ring homomorphism is injective iff its kernel is zero | Proposition | Associative rings and algebras |
| Continuum hypothesis | Axiom | Mathematical logic and foundations |
| Derivative as a little-o approximation | Proposition | Real functions |
| Diagonalisable iff the minimal polynomial has distinct roots | Theorem | Linear and multilinear algebra; matrix theory |
| Compact Hausdorff spaces are normal | Theorem | General topology |
| Change of basis formula | Theorem | Linear and multilinear algebra; matrix theory |
| A quotient is abelian iff the subgroup contains the commutator subgroup | Proposition | Group theory and generalizations |
| Test for exactness | Theorem | Ordinary differential equations |
| First isomorphism theorem for linear maps | Theorem | Linear and multilinear algebra; matrix theory |
| Exponentials dominate powers in sequences | Proposition | Real functions |
| A series of nonnegative terms converges iff its partial sums are bounded | Proposition | Sequences, series, summability |
| Mantel's theorem | Theorem | Combinatorics |
| Open connected subsets of R^n are path-connected | Theorem | General topology |
| Compact subsets of a Hausdorff space are closed | Proposition | General topology |
| Well-ordering theorem | Theorem | Mathematical logic and foundations |
| Irrationality of square roots of non-squares | Theorem | Number theory |
| Completeness via Cauchy sequences | Proposition | General topology |
| A homomorphism is injective iff its kernel is trivial | Proposition | Group theory and generalizations |
| The rationals form a field | Theorem | Field theory and polynomials |
| Limits preserve non-strict inequalities | Proposition | Sequences, series, summability |
| Liouville's theorem on Diophantine approximation | Theorem | Number theory |
| Classification of conics by the discriminant | Theorem | Geometry |
| Degree of a simple extension is the degree of the minimal polynomial | Corollary | Field theory and polynomials |
| Mean value theorem for integrals | Theorem | Real functions |
| Ore's theorem | Theorem | Combinatorics |
| Apollonius's theorem | Theorem | Geometry |
| The integral of a continuous function is an antiderivative | Corollary | Real functions |
| Implicit function theorem for a surjective derivative | Corollary | Real functions |
| Double negation law | Theorem | Mathematical logic and foundations |
| The minimal polynomial is irreducible | Proposition | Field theory and polynomials |
| Fundamental theorem of calculus | Theorem | Real functions |
| Bayes' theorem | Theorem | Statistics |
| Variance as the mean of the square minus the square of the mean | Proposition | Probability theory and stochastic processes |
| Epsilon-net characterisation of total boundedness | Proposition | General topology |
| Trace is the sum of the eigenvalues | Theorem | Linear and multilinear algebra; matrix theory |
| Closed formula for Catalan numbers | Theorem | Combinatorics |
| Schwarz's theorem for C^2 functions | Theorem | Real functions |
