Proof of Theorem EMP

Let the vectors $\vectorlist{A}{n}$ denote the columns of $A$ and let the vectors $\vectorlist{B}{p}$ denote the columns of $B$. Then for $1\leq i\leq m$, $1\leq j\leq p$, \begin{align*} \matrixentry{AB}{ij} &=\vectorentry{A\vect{B}_j}{i}&&\knowl{./knowls/definition.MM.knowl}{\text{Definition MM}}\\ &=\vectorentry{ \vectorentry{\vect{B}_j}{1}\vect{A}_1+ \vectorentry{\vect{B}_j}{2}\vect{A}_2+ \cdots+ \vectorentry{\vect{B}_j}{n}\vect{A}_n }{i}&&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}}\\ &= \vectorentry{\vectorentry{\vect{B}_j}{1}\vect{A}_1}{i}+ \vectorentry{\vectorentry{\vect{B}_j}{2}\vect{A}_2}{i}+ \cdots+ \vectorentry{\vectorentry{\vect{B}_j}{n}\vect{A}_n}{i} &&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\ &= \vectorentry{\vect{B}_j}{1}\vectorentry{\vect{A}_1}{i}+ \vectorentry{\vect{B}_j}{2}\vectorentry{\vect{A}_2}{i}+ \cdots+ \vectorentry{\vect{B}_j}{n}\vectorentry{\vect{A}_n}{i} &&\knowl{./knowls/definition.CVSM.knowl}{\text{Definition CVSM}}\\ &= \matrixentry{B}{1j}\matrixentry{A}{i1}+ \matrixentry{B}{2j}\matrixentry{A}{i2}+ \cdots+ \matrixentry{B}{nj}\matrixentry{A}{in} &&\knowl{./knowls/definition.M.knowl}{\text{Definition M}}\\ &= \matrixentry{A}{i1}\matrixentry{B}{1j}+ \matrixentry{A}{i2}\matrixentry{B}{2j}+ \cdots+ \matrixentry{A}{in}\matrixentry{B}{nj} &&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &=\sum_{k=1}^{n}\matrixentry{A}{ik}\matrixentry{B}{kj} \end{align*}