Mathematical theorems

Establishes a relationship between the sides of a right triangle.

Every proposition is either true or false in classical logic.

Every non-constant polynomial with complex coefficients has at least one complex root.

Every integer greater than one has a unique factorization into prime numbers, up to order.

More items than containers implies some container holds at least two items.

Every nonempty set of natural numbers has a least element.

Integration and differentiation are somewhat inverse processes.

he absolute value of an inner product is bounded by the product of the norms.

Every continuous map from a compact convex set to itself has a fixed point.

Relates a line integral around a simple curve to a double integral over the region it encloses.

Generalizes the previous theorem to an arbitrary number of dimensions.

Every simple closed curve in the plane divides the plane into exactly two regions.

Complete metric spaces cannot be countable unions of nowhere dense sets.

Any sufficiently expressive formal system cannot be both complete and consistent.

If a first-order theory has an infinite model, it has models of all infinite cardinalities.