(⇒) Suppose first that $\linearsystem{A}{\vect{b}}$ is consistent. Then by Theorem CSCS, $\vect{b}\in\csp{A}$. This means that $\csp{A}=\csp{\augmented{A}{\vect{b}}}$ and so it follows that $\rank{A}=\rank{\augmented{A}{\vect{b}}}$.
(⇐) Adding a column to a matrix will only increase the size of its column space, so in all cases, $\csp{A}\subseteq\csp{\augmented{A}{\vect{b}}}$. However, if we assume that $\rank{A}=\rank{\augmented{A}{\vect{b}}}$, then by Theorem EDYES we conclude that $\csp{A}=\csp{\augmented{A}{\vect{b}}}$. Then $\vect{b}\in\csp{\augmented{A}{\vect{b}}}=\csp{A}$ so by Theorem CSCS, $\linearsystem{A}{\vect{b}}$ is consistent.