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.