Coset Product is Well-Defined/Proof 3

Proof
Let $N \lhd G$ where $G$ is a group.

Let $a, a', b, b' \in G$ such that:
 * $N \circ a = N \circ a'$

and:
 * $N \circ b = N \circ b'$

We need to show that:
 * $N \circ \paren {a \circ b} = N \circ \paren {a' \circ b'}$

So: