[Community Question] Linear-algebra: Singular value inequality for sum of 2 matrices

One of our user asked:

I found a theorem mentioned in a couple of places, but could not find a proof. The theorem states the following:

Let $A, B \in \mathbb{F^{m,n}}$, $p=min(m,n)$ with singular values $\sigma_1(A) \geqslant...\geqslant \sigma_p(A)$ and $\sigma_i(B) \geqslant...\geqslant \sigma_p(B)$ respectively, then $\sigma_{i+j-1}(A+B) \leqslant \sigma_i(A) + \sigma_j(B)$.

I am looking for a proof of the above. Thanks in advance.

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