Consider the following proof that if x and y are two even integers, then their sum is even.

Proof: Assume that x and y are two even integers. By definition, there are some integers p and q such that x $=2$p$,$ and y $=2$q$.$ Let k $=$ p $+$ q be an integer, then x $+$ y $=2$p $+2$q $=2($p $+$ q$)=2$k$.$ By definition, x $+$ y is even.