I am having difficulty in understanding the proof of Corollary 3 on Pg. 44 of Miles Reid's Undergraduate Commutative Algebra
Corollary 3 Let $(A,m)$ be a local ring, $M$ an $A$-module, and $N \subset M$ a submodule; suppose that $M/N$ is finite over $A$, and that $M = N+mM$; then $N = M$.
Proof Since $m(M/N)$ = $M/N$, ...
Taking quotient by $N$ on both sides in $M = N+mM$, we get $M/N = (N+mM)/N = mM/(N \cap mM)$, but I can't see how this is $m(M/N)$. Maybe I am missing something easy here, but could someone elaborate on this?