Interchange of a Finite Double Sum

Let $K$ be a field, let $m,n$ be natural numbers, and let $[m]$ and $[n]$ be the initial segments they determine. Let $a\in(K^{n})^{m}$ be an $m$-tuple of $n$-tuples in $K$, with components $(a_{j})_{k}$ for $j\in[m]$ and $k\in[n]$, and let $b\in K^{m}$ and $c\in K^{n}$ be given by the finite sums

$$b_{j}=\sum_{k=1}^{n}(a_{j})_{k},\qquad c_{k}=\sum_{j=1}^{m}(a_{j})_{k}.$$

Then

$$\sum_{j=1}^{m}b_{j}=\sum_{k=1}^{n}c_{k},$$

written in abbreviated form as

$$\sum_{j=1}^{m}\sum_{k=1}^{n}(a_{j})_{k}=\sum_{k=1}^{n}\sum_{j=1}^{m}(a_{j})_{k}.$$