From the solution to Exercise CRS.T40 we know that $\csp{AB}\subseteq\csp{A}$. So to establish the set equality (Definition SE) we need to show that $\csp{A}\subseteq\csp{AB}$.
Choose $\vect{x}\in\csp{A}$. By Theorem CSCS the linear system $\linearsystem{A}{\vect{x}}$ is consistent, so let $\vect{y}$ be one such solution. Because $B$ is nonsingular, and linear system using $B$ as a coefficient matrix will have a solution (Theorem NMUS). Let $\vect{w}$ be the unique solution to the linear system $\linearsystem{B}{\vect{y}}$. All set, here we go,
\begin{align*}
\left(AB\right)\vect{w}
&=A\left(B\vect{w}\right)&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\
&=A\vect{y}&&\text{$\vect{w}$ solution to $\linearsystem{B}{\vect{y}}$}\\
&=\vect{x}&&\text{$\vect{y}$ solution to $\linearsystem{A}{\vect{x}}$}\\
\end{align*}
This says that the linear system $\linearsystem{AB}{\vect{x}}$ is consistent, so by Theorem CSCS, $\vect{x}\in\csp{AB}$. So $\csp{A}\subseteq\csp{AB}$.