Computing SVD using eigendecomposition of $A^TA$ and $AA^T$

Question

From Matrix Computations 4th Ed, Golub, Van Loan, pg. 489

$A \in \mathbb{R}^{mxn}$ with $m >= n$

1. Form C = $A^TA$
2. Use the symmetric QR algorithm to compute $V_1^TCV_1 = diag(\sigma_i^2)$
3. Apply QR with column pivoting to $AV_1$ obtaining $U^T(AV_1)\Pi=R$

Since R has orthogonal columns, it follows that $U^TA(V_1\Pi)$ is diagonal.

My question is: how can we deduce that R has orthogonal columns? Also, I'm guessing that any upper-triangular matrix with orthogonal columns must be diagonal.

Thanks!