This is copied from a Facebook chat message I had with someone a few weeks ago, with wordpress latex applied over the math:
A couple weeks ago, I learned the statement of the second isomorphism theorem, which states that given a subgroup and normal subgroup of , is a subgroup of and is a normal subgroup of , with isomorphic to .
Any element of can be represented as for , where the on the LHS is in . A similar statement of representation via , holds for . Define with , which is bijective. By normality, . Thus, is an isomorphism. QED.