# Corollaries

Reference > Mathematics > Introduction to ProofsIn the previous section we talked about lemmas, and I said that sometimes a lemma is called a "pre-proof" - it's something you need to prove in order to complete a bigger proof. A Corollary could be described as a "post-proof." A corollary is something that follows almost obviously from a theorem you've proved. You work to prove something, and when you're all done, you realize, "Oh my goodness! If this is true, than [another proposition] must also be true!"

Here's an example: In one of the early sections of this unit we proved something we called the "Sum and Product Theorem." It looked like this:

Sum and Product Theorem: x^{2} + (a + b)x + ab = (x + a)(x + b).

Later on, in the problems section of that page, we asked the student to prove the following:

a^{2} - b^{2} = (a - b)(a + b)

This can be done by repeated use of the distributive property, followed by the transitive property, but there is a quicker way to solve it, based on the Sum and Product Theorem. And since our proof is based on the Sum and Product Theorem, we could call it a corollary:

Sum and Product Corollary: a^{2} - b^{2} = (a - b)(a + b)

Here is the very simple proof:

STATEMENT REASON 1. Let y = a^{2}+ (b - b)a + b(-b) Variable assignment 2. y = (a + b)(a + -b) Sum and Product Theorem 3. y = a^{2}- b^{2}Combining Like Terms (1) 4. a^{2}- b^{2}= (a + b)(a - b) Transitive property (2,3)

The trick was in recognizing that the expression a^{2} - b^{2} could be rewritten as a "sum and product" quadratic. Once you've done that, it's easy to see that the corollary is true.

So when do you call something a corollary, and when do you call it a theorem? That's a tough call; the notion is that a corollary is easy to prove based on a single theorem, but how easy does it have to be? A three line proof? A five line proof? There's no "hard and fast" rule for that. Fortunately, unless you're doing original research, you don't have to make that decision. And if you are doing original research, you'll get to decide for yourself whether you'll call it a corollary or a theorem.

In this case, we could have just as easily called this the "Difference of Squares Theorem," and that would have been perfectly acceptable. The point here was merely to introduce you to concept, so you'll recognize what a corollary is when you see one someday down the road!