The proof of this theorem is as much about a change in notation as it is about making logical deductions. Write the system of equations $\linearsystem{A}{\vect{b}}$ as \begin{align*} a_{11}x_1+a_{12}x_2+a_{13}x_3+\dots+a_{1n}x_n&=b_1\\ a_{21}x_1+a_{22}x_2+a_{23}x_3+\dots+a_{2n}x_n&=b_2\\ a_{31}x_1+a_{32}x_2+a_{33}x_3+\dots+a_{3n}x_n&=b_3\\ \vdots&\\ a_{m1}x_1+a_{m2}x_2+a_{m3}x_3+\dots+a_{mn}x_n&=b_m \end{align*}
Notice then that the entry of the coefficient matrix $A$ in row $i$ and column $j$ has two names: $a_{ij}$ as the coefficient of $x_j$ in equation $i$ of the system and $\vectorentry{\vect{A}_j}{i}$ as the $i$-th entry of the column vector in column $j$ of the coefficient matrix $A$. Likewise, entry $i$ of $\vect{b}$ has two names: $b_i$ from the linear system and $\vectorentry{\vect{b}}{i}$ as an entry of a vector. Our theorem is an equivalence (Proof Technique E) so we need to prove both “directions.”
(⇐) Suppose we have the vector equality between $\vect{b}$ and the linear combination of the columns of $A$. Then for $1\leq i\leq m$, \begin{align*} b_i &=\vectorentry{\vect{b}}{i}&&\knowl{./knowls/definition.CV.knowl}{\text{Definition CV}}\\ &=\vectorentry{ \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 }{i}&&\text{Hypothesis}\\ &= \vectorentry{\vectorentry{\vect{x}}{1}\vect{A}_1}{i}+ \vectorentry{\vectorentry{\vect{x}}{2}\vect{A}_2}{i}+ \vectorentry{\vectorentry{\vect{x}}{3}\vect{A}_3}{i}+ \cdots+ \vectorentry{\vectorentry{\vect{x}}{n}\vect{A}_n}{i} &&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}}\\ &= \vectorentry{\vect{x}}{1}\vectorentry{\vect{A}_1}{i}+ \vectorentry{\vect{x}}{2}\vectorentry{\vect{A}_2}{i}+ \vectorentry{\vect{x}}{3}\vectorentry{\vect{A}_3}{i}+ \cdots+ \vectorentry{\vect{x}}{n}\vectorentry{\vect{A}_n}{i} &&\knowl{./knowls/definition.CVSM.knowl}{\text{Definition CVSM}}\\ &= \vectorentry{\vect{x}}{1}a_{i1}+ \vectorentry{\vect{x}}{2}a_{i2}+ \vectorentry{\vect{x}}{3}a_{i3}+ \cdots+ \vectorentry{\vect{x}}{n}a_{in} &&\knowl{./knowls/definition.CV.knowl}{\text{Definition CV}}\\ &= a_{i1}\vectorentry{\vect{x}}{1}+ a_{i2}\vectorentry{\vect{x}}{2}+ a_{i3}\vectorentry{\vect{x}}{3}+ \cdots+ a_{in}\vectorentry{\vect{x}}{n} &&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}} \end{align*}
This says that the entries of $\vect{x}$ form a solution to equation $i$ of $\linearsystem{A}{\vect{b}}$ for all $1\leq i\leq m$, in other words, $\vect{x}$ is a solution to $\linearsystem{A}{\vect{b}}$.
(⇒) Suppose now that $\vect{x}$ is a solution to the linear system $\linearsystem{A}{\vect{b}}$. Then for all $1\leq i\leq m$, \begin{align*} \vectorentry{\vect{b}}{i} &=b_i&&\knowl{./knowls/definition.CV.knowl}{\text{Definition CV}}\\ &= a_{i1}\vectorentry{\vect{x}}{1}+ a_{i2}\vectorentry{\vect{x}}{2}+ a_{i3}\vectorentry{\vect{x}}{3}+ \cdots+ a_{in}\vectorentry{\vect{x}}{n} &&\text{Hypothesis}\\ &= \vectorentry{\vect{x}}{1}a_{i1}+ \vectorentry{\vect{x}}{2}a_{i2}+ \vectorentry{\vect{x}}{3}a_{i3}+ \cdots+ \vectorentry{\vect{x}}{n}a_{in} &&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &= \vectorentry{\vect{x}}{1}\vectorentry{\vect{A}_1}{i}+ \vectorentry{\vect{x}}{2}\vectorentry{\vect{A}_2}{i}+ \vectorentry{\vect{x}}{3}\vectorentry{\vect{A}_3}{i}+ \cdots+ \vectorentry{\vect{x}}{n}\vectorentry{\vect{A}_n}{i} &&\knowl{./knowls/definition.CV.knowl}{\text{Definition CV}}\\ &= \vectorentry{\vectorentry{\vect{x}}{1}\vect{A}_1}{i}+ \vectorentry{\vectorentry{\vect{x}}{2}\vect{A}_2}{i}+ \vectorentry{\vectorentry{\vect{x}}{3}\vect{A}_3}{i}+ \cdots+ \vectorentry{\vectorentry{\vect{x}}{n}\vect{A}_n}{i} &&\knowl{./knowls/definition.CVSM.knowl}{\text{Definition CVSM}}\\ &=\vectorentry{ \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 }{i}&&\knowl{./knowls/definition.CVA.knowl}{\text{Definition CVA}} \end{align*} So the entries of the vector $\vect{b}$, and the entries of the vector that is the linear combination of the columns of $A$, agree for all $1\leq i\leq m$. By Definition CVE we see that the two vectors are equal, as desired.