Proposition. The sum of two odd integers is even.

We show that if x and y are odd integers, then x + y is an even integer. Let x and y be odd integers. By Definition of odd, we know that there are integers a and b with x = 2a + 1 and y = 2b + 1. Observe that x + y = (2a + 1) + (2b + 1) = 2a + 2b + 2 =2(a + b + 1). Therefore there is an integer c, namely a + b + 1, so that x + y = 2c. Hence 2|(x + y) and therefore by definition of even, x + y is even.