$A$ is nonsingular, so by Theorem NMRRI there is a sequence of row operations that will convert $A$ into $I_n$. It is this same sequence of row operations that will convert $M$ into $N$, since having the identity matrix in the first $n$ columns of $N$ is sufficient to guarantee that $N$ is in reduced row-echelon form.
If we consider the systems of linear equations, $\linearsystem{A}{\vect{e}_i}$, $1\leq i\leq n$, we see that the aforementioned sequence of row operations will also bring the augmented matrix of each of these systems into reduced row-echelon form. Furthermore, the unique solution to $\linearsystem{A}{\vect{e}_i}$ appears in column $n+1$ of the row-reduced augmented matrix of the system and is identical to column $n+i$ of $N$. Let $\vectorlist{N}{2n}$ denote the columns of $N$. So we find, \begin{align*} AJ=&A[\vect{N}_{n+1}|\vect{N}_{n+2}|\vect{N}_{n+3}|\ldots|\vect{N}_{n+n}]\\ =&[A\vect{N}_{n+1}|A\vect{N}_{n+2}|A\vect{N}_{n+3}|\ldots|A\vect{N}_{n+n}]&&\knowl{./knowls/definition.MM.knowl}{\text{Definition MM}}\\ =&[\vect{e}_1|\vect{e}_2|\vect{e}_3|\ldots|\vect{e}_n]\\ =&I_n&&\knowl{./knowls/definition.IM.knowl}{\text{Definition IM}} \end{align*} as desired.