Suppose matrix $A$ is an $n \times n$ orthogonal matrix and $S=\{\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_n\}$ where $\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_n$ are orthogonal to each other.
Now, since both $A$ and $S$ are orthogonal, by definition, I know that the matrix $T=AS=\begin{bmatrix} A\vec{u}_1 & A\vec{u}_2 & \cdots & A\vec{u}_n \end{bmatrix}$ is also orthogonal. But why is $T$ orthogonal too?
I tried to do this to prove that $T$ is orthogonal: $\begin{align*} T&=AS\\ T^TT&=(AS)^TAS\\ T^TT&=S^TA^TAS \end{align*}$ At this point, I am stuck because $A^TA$ don't give me the identity matrix unless $A$ is orthonormal, which it isn't ($A$ is only orthogonal). This goes the same for $S$. How should I move on from here to show that $T$ is indeed orthogonal?