Proof of Theorem TT

We again want to prove an equality of matrices, so we work entry-by-entry and use Definition ME. For $1\leq i\leq m$, $1\leq j\leq n$, \begin{align*} \matrixentry{\transpose{\left(\transpose{A}\right)}}{ij} &=\matrixentry{\transpose{A}}{ji}&&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}}\\ &=\matrixentry{A}{ij}&&\knowl{./knowls/definition.TM.knowl}{\text{Definition TM}} \end{align*}

Since the matrices $\transpose{\left(\transpose{A}\right)}$ and $A$ agree at each entry, Definition ME tells us the two matrices are equal.