In mathematics or logic, a statement whose validity has been established or proved.
It consists of a hypothesis and a conclusion, beginning with certain assumptions that are necessary and sufficient to establish a result. A system of theorems that build on and augment each other constitutes a theory. Within any theory, however, only statements that are essential, important, or of special interest are called theorems. Less important statements, usually stepping-stones in proof s of more important results, are called lemmas. A statement proved as a direct consequence of a theorem is a corollary of the theorem. Some theorems (and even lemmas and corollaries) are singled out and given titles (e.g., Gödel's theorem , fundamental theorem of algebra , fundamental theorem of calculus , Pythagorean theorem ).