Minkowski's Inequality for Sums/Equality

Theorem
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \R_{\ge 0}$ be non-negative real numbers.

Let $p \in \R$, $p \ne 0$ be a real number.

Then equality in Minkowski's Inequality for Sums, that is:
 * $\ds \paren {\sum_{k \mathop = 1}^n \paren {a_k + b_k}^p}^{1 / p} = \paren {\sum_{k \mathop = 1}^n a_k^p}^{1 / p} + \paren {\sum_{k \mathop = 1}^n b_k^p}^{1 / p}$

holds, for all $k \in \closedint 1 n$:
 * $\dfrac {a_k} {b_k} = c$

for some $c \in \R_{>0}$.