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 5 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 3a+(3b+1)+(3c+2)=3(a+b+c)+3, which is a multiple of 3. Q.E.D.