When the phrase "without loss of generality" is used in a proof, we are asserting that by proving one case of a theorem, no additional argument is required to prove other specified cases.
Show that if and are integers and both and are even, then both and are even.
We will prove using contraposition, the notion of WLOG, and proof by cases. Suppose that and are not both even. That is, assume is odd or is odd or both. WLOG, we assume is odd, so that for some integer .
To complete the proof, we need to show that is odd or is odd. Consider two cases: () is even, and () is odd. In (), for some integer , so that is odd. In (), for some integer , so that is odd. This completes the proof by contraposition. (Note that our use of without loss of generality within the proof is justified because the proof when is odd can be obtained by simply interchanging the roles of and in the proof we have given.)