Prove that the polar decomposition of normal matrices, $A=SU$, is such that $SU=US$

Let $\,A=U|A|$, then $\,A^*=|A|U^*$. By normality one obtains $$U|A|^2U^* = AA^* = A^*A = |A|^2,$$ an equality of positive-semidefinite matrices.

"Positive square-rooting" yields $\,U|A|U^* = |A|\;\Longleftrightarrow\; U|A| = |A|U$.