How to prove that $U$ and $V$ are orthogonal?. If, $MM^TU=U\Sigma^2$ and $M^TMV=V\Sigma^2$ then

Question

Let $M \in R^{m×n}$. In the context of the SVD theory, how to prove that $U$ and $V$ are orthogonal , If $U$ and $V$ are invertible and, $MM^TU=U\Sigma^2$, $M^TMV=V\Sigma^2$.

Orthogonality definition: If $U$ and $V$ are orthogonal then $U^{-1}=U^t$ and $V^{-1}=V^t$.