For example, this statement can be solved with a direct proof: "if x and y are even integers, then x+y is an even integer." Since x and y are even, then we can say that "x=2m" and "y=2n," where "m" and "n" are integers. (This is a lemma.) We can also say that "m+n" is an integer, because adding two integers gives an integer. (This is another lemma.) Then we can say that "x+y=2m+2n=2(m+n)". Since "m+n" is an integer, we can say "2(m+n)=2k", where "k=m+n". We know that any integer times 2 is an even integer. We can then say that any two even integers added together give an even integer.