Theorems are the need and special concern of mathematics and when talking about them, reference is made to those statements that can be shown to be true within a logical framework.
Generally, the theorems are composed of a number of conditions that can be listed or anticipated in advance to which they are called responses. Following these, the conclusion or mathematical statement will appear, which will obviously always be true in the conditions of the work in question, that is, first of all in the informative content of the theorem, what will be established is the relationship that exists between the hypothesis and the thesis or completion of work.
But there is something unavoidable for mathematics when a certain statement is plausible to become a theorem and that is that it must be interesting enough within and for the mathematical community, otherwise and unfortunately, it can simply be a motto, a corollary or simply a proposition, never being able to become a theorem.
And in order to clarify the issue a little more, it is also necessary to distinguish the concepts that we mentioned above, so that, even if we are not part of a mathematical community, we can recognize when it is a theorem, a lemma, a corollary or a proposition.
A Lemma is a proposition, but it is part of a longer theorem. The Corollary for its part is a statement that follows a theorem and finally the proposition is a result that is not associated with any particular theorem.
At the beginning we indicated that a theorem is a statement that can only be proved within a logical framework, while with a logical framework we refer to a set of axioms or axiomatic system and an inference process that will allow us to derive theorems from of the axioms and theorems that have already been derived previously.
On the other hand, the finite sequence of well-formed logical formulas will be called a proof of this theorem.
Although not with the special attention that mathematics devotes to theorems, disciplines such as physics or economics usually produce statements that are deduced from others and that are also called theorems.
