 The next mathematical operation we'll look at will be subtraction, and we define subtraction in a very specific way. The definition is going to be based on addition, and it relies on the relationship between two facts. Suppose that I know that A plus B is equal to C, so I know an addition fact. What's nice is that I immediately know a subtraction fact, which is that A is equal to C minus B, and conversely, if A is equal to C minus B, then I know that A plus B is equal to C. And so here's everything that we need to know about subtraction, and it really comes down to, if I know how to add, I immediately know how to subtract. Now, we'll actually have to do a little bit more than that, but for now, the thing to notice is that we have this very nice definition for what a subtraction is, and again in line with earlier problems, I may have called to prove something, and the thing to notice here is that my addition here, A plus B is equal to C, is part of my definition of subtraction. Which means that when I go to prove a subtraction statement, I can assume the addition, and I don't have to say too much about it. For example, let's try and prove that 5 minus 3 is equal to 2. And again, this is a prove statement, and again in general, what we do when we have a prove statement, we want to rely on our definitions and theorems, and not on any actual method of performing the subtraction. Again, we know we're told 5 minus 3 equals 2. We're never going to be asked to prove something that isn't true, or something we don't believe is true, so we already believe 5 minus 3 is equal to 2. This is not a question of finding 5 minus 3, but it's rather proving that when we do find it, we do have the correct value. And again, if this seems to be rather pointless, the thing to remember is that the purpose of a proof is not to establish the truth, or falsehood in some cases, of a particular statement. It's not that anybody has any doubt that 5 minus 3 is equal to 2. The purpose of a proof is to remind us of things that we already know, or things that we should know. So again, we can very often in this course rely on definitions, and so let's pull up that definition of subtraction. Again, here it is, so suppose a plus b is equal to c, then a is equal to c minus b, and conversely. And I'll compare the definition that we have with the subtraction. We want to prove 5 minus 3 equals to 2, and by comparison of the two statements, I see that c is 5, b is 3, and a is 2. So I can fill those in. I'll instantiate the proof here, and I'll go ahead and substitute those in since a plus b equals c. 2 plus 3 equals 5, there's the first part of my proof, then 5 minus 3 equals 2. And again, because the addition is part of the definition of the subtraction, then the definition of subtraction tells me that I don't have to worry about why I know 2 plus 3 is equal to 5. I can start with 2 plus 3 equals 5, make sure it's a true statement, and then immediately conclude 5 minus 3 is equal to 2. And again, as an answer to a question that asks you to prove something, what you have to show is everything in green. So there's the tie-in between our definition of subtraction and the specific statement we want to prove, and there's our proof.