Proof of Theorem NMPEM

Since $A$ is nonsingular, it is row-equivalent to the identity matrix by Theorem NMRRI, so there is a sequence of $t$ row operations that converts $I$ to $A$. For each of these row operations, form the associated elementary matrix from Theorem EMDRO and denote these matrices by $E_1,\,E_2,\,E_3,\,\dots,\,E_t$. Applying the first row operation to $I$ yields the matrix $E_1I$. The second row operation yields $E_2(E_1I)$, and the third row operation creates $E_3E_2E_1I$. The result of the full sequence of $t$ row operations will yield $A$, so \begin{equation*} A= E_t\dots E_3E_2E_1I= E_t\dots E_3E_2E_1 \end{equation*}

Other than the cosmetic matter of re-indexing these elementary matrices in the opposite order, this is the desired result.