 So let's try to prove the commutativity of addition. Now, we've already shown that zero commutes for any natural number n plus zero is equal to zero plus n. To prove commutativity in general, we want to show that n plus one is equal to one plus n, n plus two is equal to two plus n, n plus three is equal to three plus n, and so on. And so it seems we want an induction proof. So let's try it. So an induction proof, we have to prove the base step. So we want to prove that n plus zero is equal to zero plus n, and we've already done that. So let's prove our induction step. So suppose n plus k is equal to k plus n. We'd like to show that this statement also holds for n plus k star. So let's consider the value of n plus k star and also the value of k star plus n. And by this point, you know the drill definitions are the whole of mathematics. All else is commentary. And n plus k star, well, from our definition of addition, that's the same as the successor of n plus k. And we proved earlier that the successor of n is the same as n plus one. So the successor of n plus k is n plus k plus one. And, well, we do know that's k plus n plus one. Let's see if we can figure out what k star plus n is. So we know that n star is n plus one. So k star is k plus one. And then we can, oh, we can group this one plus n, and then that's n plus one, and then we can regroup it, and there's our proof. And remember, if you don't find the flaws in your argument, someone else will. And we should look at this proof very carefully. This first step here, we regrouped k plus one plus n as k plus one plus n, and that's the associative property of addition, which we haven't proved. And so it seems that we need to prove associativity first. So if I want to prove the associative property, this is really the same as proving a whole bunch of statements. And this is the type of thing that we'll prove by induction. While we could start by proving a plus b plus zero is a plus b plus zero, that's too easy. By this point, you should pretty much be able to write out a proof without any difficulty. So in line with the idea of you should prove things multiple times, challenge yourself. And so in this case, let's make the actual base step this first claim. So let's consider a plus b plus one and compare it to a plus b plus one. So again, we've proved the theorem that the successor of n is n plus one, which means that a plus b plus one is the successor of a plus b. And similarly, a plus b plus one, well, that's the same as a plus the successor of b. Definitions are the whole of mathematics. All else is commentary. Our definition of addition says that a plus the successor of b is the same as the successor of a plus b. And so a plus b plus one is the same as a plus b plus one. And let's consider our induction step. So again, suppose that our statement is true for k. We want to show that it's true for k star. So let's compare a plus b plus k star to a plus b plus k star. Definitions are the whole of mathematics. All else is commentary. When I add the successor, it's the same as the successor of the sum. And our option that the statement is true for k means that this is the successor of a plus b plus k. And again, when you add a successor, it's the same as the successor of the sum. And so we have our proof of the induction step. Namely that a plus b plus k star and a plus b plus k star are the same thing. And so we have associativity. And so we're allowed to make this regrouping. And again, if you don't find the flaws in your argument, someone else will. Remember, we're trying to prove commutativity. But here, we used the commutativity of 1 and n. And so it seems we need to prove that n plus 1 is the same as 1 plus n. And so we should prove this by, well again, an induction proof seems appropriate. So we want to prove our base step, 0 plus 1 equals 1 plus 0. Well, we've already done that because we prove that 0 does actually commute. And our induction step, suppose our statement is true for k and that k plus 1 is equal to 1 plus k. Our goal is to show that k star plus 1 is equal to 1 plus k star. And definitions are the whole of mathematics. All else is commentary. So let's see. 1 plus k star, we can go back a step from our definition of addition. That's the successor of 1 plus k. We've assumed that 1 commutes with k. So 1 plus k is the same as k plus 1. That takes us back one further step. So definitions are the whole of mathematics. All else is commentary. But sometimes that commentary is useful and will save us a couple of steps. And so one of those commentaries is a theorem for all n. We know that n star is the same as n plus 1. So we know that k plus 1 star is the same as k plus 1 plus 1. And in fact, that connects our final bridge because k plus 1 is k star. And so we have our first line and our starting point. Now, remember that you should be able to drive over the bridge forward. So let's just make sure that that can happen. So suppose k plus 1 equals 1 plus k, k star plus 1. Our theorem says that k star is k plus 1. So that's our first step. Our theorem says that k plus 1 plus 1 is k plus 1 star. So that's our second step. Our assumption is that k plus 1 is equal to 1 plus k. So that's our third step. And our definition of addition says that 1 plus k successor is 1 plus the successor of k. And now we can prove commutativity. So again, we prove the base step. Then we tried to prove the result for the successor. But in order to do that, we needed associativity. And so we proved associativity, which allowed us to regroup the 1 plus n. But then we needed to prove that 1 plus n was equal to n plus 1. So we proved n plus 1 equals 1 plus n, which allowed us to reverse. Associativity allows us to regroup. And so n plus k star is equal to k star plus n. And that proves our induction step and gives us our proof of commutativity. Now it's worth noting that while we could have simply claimed a plus b is equal to b plus a, and we often do that in lower level courses, our proof led us to introduce and prove the associative property. Proof is a way of discovering things you might not have known about. It also caused us to review our definition of addition and reminded us of the theorems we've already proven. And proof is a good way of studying mathematics. And in proof, the thing to remember is that it's the journey, not the destination.