Proposition. The sum of two odd integers is even.
1. We show that if x and y are odd integers then x + y is an even integer
.2. Let x and y be odd integers.
3. Since x is odd, we know by Definition 1.4 that there is an integer a with x = 2a + 1.
4. Likewise, since y is odd, there is an integer b with y = 2b + 1.
5. Observe that x + y = (2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1).
6. Therefore there is an integer c (namely a + b + 1) with x + y = 2c.
7. Therefore by Definition 1.2, 2|(x + y).
8. Hence (Definition 1.1) x + y is even.