Proof of Theorem SLEMM

This theorem says that two sets (of solutions) are equal. So we need to show that one set of solutions is a subset of the other, and vice versa (Definition SE). Let $\vectorlist{A}{n}$ be the columns of $A$. Both of these set inclusions then follow from the following chain of equivalences (Proof Technique E), \begin{align*} &\vect{x}\text{ is a solution to }\linearsystem{A}{\vect{b}}\\ &\iff \vectorentry{\vect{x}}{1}\vect{A}_1+ \vectorentry{\vect{x}}{2}\vect{A}_2+ \vectorentry{\vect{x}}{3}\vect{A}_3+ \cdots+ \vectorentry{\vect{x}}{n}\vect{A}_n =\vect{b}&&\knowl{./knowls/theorem.SLSLC.knowl}{\text{Theorem SLSLC}}\\ &\iff \vect{x}\text{ is a solution to }A\vect{x}=\vect{b}&&\knowl{./knowls/definition.MVP.knowl}{\text{Definition MVP}} \end{align*}