What are the differences between theorems, corollaries, propositions, ?

A proposition is a statement which is offered up for investigation as to its truth or falsehood. The term theorem is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch. A corollary is a proof which is a direct result, or a direct application, of another proof.It can be considered as being a proof for free on the back of a proof which has been paid for with blood, sweat and tears.The word is ultimately derived from the Latin corolla, meaning small garland, or the money paid for it. Hence has the sense something extra.