Proof of Theorem CCM

For $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{\conjugate{\left(\conjugate{A}\right)}}{ij} &=\conjugate{\matrixentry{\conjugate{A}}{ij}}&&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\conjugate{\conjugate{\matrixentry{A}{ij}}}&&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\matrixentry{A}{ij}&&\knowl{./knowls/theorem.CCT.knowl}{\text{Theorem CCT}}% \end{align*}

Since the matrices $\conjugate{\left(\conjugate{A}\right)}$ and $A$ are equal in each entry, Definition ME says that $\conjugate{\left(\conjugate{A}\right)}=A$.