Without loss of generality (abbreviated as w.l.o.g.) is a common idiom in mathematics that remarks that we can introduce a new assumption reducing the proof to a special case, and the proof for the other cases either follows from the special case, can be reasoned in an analogous way, or is trivial.

wlog is tightly related to case exhaustion.

Example with reduction to a special case

Example with analogous reasoning

Example with triviality

Theorem: In every set of natural numbers there are three numbers which sum a multiple of $3$ .

Proof:

w.l.o.g. assume that there are no three numbers with the same residue modulo $3$ in the set. Otherwise, the sum of those three numbers is a multiple of $3$ .

Now, there are $5$ numbers and $3$ possible residues, so at least there is one number for each residue (otherwise, there could be a maximum of $2$ residue classes times a maximum of $2$ number per class, for a total of $4$ numbers). But $3 a + (3 b + 1) + (3 c + 2) = 3 (a + b + c) + 3$ , which is a multiple of $3$ . Q.E.D.