We will work with the vector equality representations of the relevant systems of equations, as described by Theorem SLEMM.
(⇐) Suppose $\vect{y}=\vect{w}+\vect{z}$ and $\vect{z}\in\nsp{A}$. Then
\begin{align*}
A\vect{y}&=A(\vect{w}+\vect{z})&&\text{Substitution}\\
&=A\vect{w}+A\vect{z}&&\knowl{./knowls/theorem.MMDAA.knowl}{\text{Theorem MMDAA}}\\
&=\vect{b}+\zerovector&&\vect{z}\in\nsp{A}\\
&=\vect{b}&&\knowl{./knowls/property.ZC.knowl}{\text{Property ZC}}\\
\end{align*}
demonstrating that $\vect{y}$ is a solution.
(⇒) Suppose $\vect{y}$ is a solution to $\linearsystem{A}{\vect{b}}$. Then
\begin{align*}
A(\vect{y}-\vect{w})
&=A\vect{y}-A\vect{w}&&\knowl{./knowls/theorem.MMDAA.knowl}{\text{Theorem MMDAA}}\\
&=\vect{b}-\vect{b}&&\text{$\vect{y},\,\vect{w}$ solutions to $A\vect{x}=\vect{b}$}\\
&=\zerovector&&\knowl{./knowls/property.AIC.knowl}{\text{Property AIC}}\\
\end{align*}
which says that $\vect{y}-\vect{w}\in\nsp{A}$. In other words, $\vect{y}-\vect{w}=\vect{z}$ for some vector $\vect{z}\in\nsp{A}$. Rewritten, this is
$\vect{y}=\vect{w}+\vect{z}$, as desired.