 Let's do another example. We want to prove the following. The intersection of A's, or let's say A's intersection with the union of B and C equals the union of A's intersection with B and A's intersection with C. What do we pull out of our box? Something that we've proven before or that we have to find. We know what the intersection is or a set of all elements such that X is an element of the first set and X is an element of the second set. The union we've defined as the set of all elements, X such that X is an element of the first one or X is an element of the second one. And we say that two sets are equal to each other if and only if one is a subset. First is a subset of the second and the second is a subset of the first. That's all we're going to do. So let's start off. Remember, this is an equality here. We've got to prove it in both directions. One implies the other that one implies this one. So let's start off with by saying let X be an element of A's intersection with B's union of C. If that is so, what does that mean? It means I can unpack that. If X is an element of A then by our definition, X is an element of A and X is an element of B's union with C. That's what I have. By my definition it means that and then X is an element of A and X is an element of B or X is an element of C. So read my English statement here, my language statement. X is an element of A and X is an element of B or X is an element of C. Now let's suggest that. Let's take that. Let's use one of the scenarios we're going to see. If X is an element of A, so it's an element of A and B and B or C. So it's an element of A and B or it's an element of A and C. So let X be an element of A and X be an element of B. Then we have by definition X is an element of A's intersection with B or if X is an element of A, remember X is an element of A and B or X is an element of A or C. That's what my English statement there says. So if X is an element of A and X is an element of C, then by our definitions X is an element of A's intersection with C. But it can be one or the other. X is an element of A and X is an element of B or C. It's an element of A and either B or C. So it's such that X is an element of this first one A's intersection of B or X is an element of A's intersection with C. And by our definition we have then that X is an element of A's intersection with B's union of A's intersection with C. What have we proven of X is an element of that and it turns out to be an element of this. It means that we have shown that A's intersection of B's union with C is a subset of the union of A's intersection with B union intersection of C. So that was one way. Conversely what can we have? Let X be an element of this. A's intersection of B's union A intersection with C. Then what do we have? We have that X is an element of A's intersection with B or X is an element of A's intersection with C. So if that is so, then furthermore by our following definitions then we have that X is an element of A or X is an element of B I should say and. Let's have that and or we have that X is an element of A so keep on writing that today and X is an element of C. So I've just unpacked by my definition. Now let's just have a look at it. X is an element of A and X is an element of B or X is an element of A and X is an element of C. So no matter what X is an element of A, no matter either of these it's going to be an element of B or C. If X is an element of A, an element of A and then we're going to have if X is an element of B or C so X is going to be an element of A regardless because it's O. It's going to be an element of A or then it's going to be an element of B and C of B and let's see then by our definition then by our definition X is an element of B is union with C but we still have that X is an element of A on this side and X is an element of that therefore by our definitions X is an element of A's intersection with B's union of C so if that is true it means this must be a subset of that so that means that A's intersection with B's union with A's intersection of C is a subset of A's intersection of the union of B and C and since I have these two I have now got equality so I've proven that my understanding that A's intersection with B's union with C is nothing other than A's intersection with B's union of the intersection of A of A's intersection with C. QED we've proven so all we've done is we've used what we've had before we've unpacked it and we've shown it in both directions and that completes our nice easy proof