Proof of Theorem MCT

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

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