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 .
Proof:
w.l.o.g. assume that there are no three numbers with the same residue modulo in the set. Otherwise, the sum of those three numbers is a multiple of .
Now, there are numbers and possible residues, so at least there is one number for each residue (otherwise, there could be a maximum of residue classes times a maximum of number per class, for a total of numbers). But , which is a multiple of . Q.E.D.