I don't know whether you would count this, but the proof of the existence of quotient groups seems to fit your description. Some people define the product of the cosets $gH$ and $g'H$ to be the coset $gg'H$, and then go on to prove that this is well-defined. Another approach is to define two elements $g$ and $h$ of $G$ to be equivalent if $gh^{-1}\in H$, to define cosets to be equivalence classes, and to define the product of two cosets to be ... their product. That is, if $A$ and $B$ are cosets then their product is $AB=\{ab:a\in A, b\in B\}$. Of course, we have to prove that that is a coset. But if $ac^{-1}\in H$ and $bd^{-1}\in H$, then $cbd^{-1}c^{-1}\in H$ so $ab(cd)^{-1}\in H$. Thus, if $a\sim c$ and $b\sim d$, then $ab\sim cd$.