Confusion over formula for finding V during SVD

Question

I understand any rectangular mxn matrix A is a factorization of the form: $$ A = U\Sigma V^{\top}$$ where U and V are orthogonal matrices and $\Sigma$ is diagonal.
Say, I have found U and $\Sigma$ and looking for V. I have been told the following $$ v_i = \frac{(A^{\top}U)_i}{\sigma_i} $$ where 0 $\leq$ i $\leq$ p and p = min(m,n)
The remaining vectors in V can then be calculated by the cross product of these vectors. When using this formula, it seems to be correct however I am unsure why. The proof I was given is as follows $$ A = U\Sigma V^{\top} \\ A^{\top}A = A^{\top}U\Sigma V^{\top} \\ I = A^{\top}U\Sigma V^{\top} \\ A^{\top}U = V\Sigma^{\top} \\ (A^{\top}U)_i = v_i\sigma_i \\ v_i = \frac{(A^{\top}U)_i}{\sigma_i} $$ However, this would only hold if A is an orthogonal matrix which is not the case in the problems I have tried.

Answer 1

It's true because $A^\top U=(U\Sigma V^\top)^\top U=(V\Sigma^\top U^\top)U=V\Sigma^\top$.