Proof of Theorem CRMA

For $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{\conjugate{A+B}}{ij} &=\conjugate{\matrixentry{A+B}{ij}}&&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\conjugate{\matrixentry{A}{ij}+\matrixentry{B}{ij}}&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}}\\ &=\conjugate{\matrixentry{A}{ij}}+\conjugate{\matrixentry{B}{ij}}&&\knowl{./knowls/theorem.CCRA.knowl}{\text{Theorem CCRA}}\\ &=\matrixentry{\conjugate{A}}{ij}+\matrixentry{\conjugate{B}}{ij}&&\knowl{./knowls/definition.CCM.knowl}{\text{Definition CCM}}\\ &=\matrixentry{\conjugate{A}+\conjugate{B}}{ij}&&\knowl{./knowls/definition.MA.knowl}{\text{Definition MA}} \end{align*}

Since the matrices $\conjugate{A+B}$ and $\conjugate{A}+\conjugate{B}$ are equal in each entry, Definition ME says that $\conjugate{A+B}=\conjugate{A}+\conjugate{B}$.