Matrix/Staircase form by elementary manipulations/Fact

Let $K$ be a field, and let $M$ denote an $m\times n$-matrix over $K$.

Then there exist

elementary row operations, a (new) numbering of the columns

$j_1,j_2,\dots ,j_n,$

and an $r\le n$ such that, in the new matrix, the columns have the form

$s_{j_k}=\left(\begin{array}{c}{b}_{1,j_k}\\ ⋮\\ b_{k,j_k}\\ 0\\ ⋮\\ 0\end{array}\right)\text{ with }{b}_{k,j_k}\ne 0\text{ for }k\le r$

and

$s_{j_k}=\left(\begin{array}{c}{b}_{1,j_k}\\ ⋮\\ b_{r,j_k}\\ 0\\ ⋮\\ 0\end{array}\right)\text{ for }k>r.$

By further elementary row operations, and by swapping of columns, the matrix can be brought to the form

$\left(\begin{array}{ccccccc}{d}_{11}& \ast & \cdots & \ast & \ast & \cdots & \ast \\ 0& d_{22}& \cdots & \ast & \ast & \cdots & \ast \\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & d_{rr}& \ast & \cdots & \ast \\ 0& 0& \cdots & 0& 0& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & 0& 0& \cdots & 0\end{array}\right)$

with

$d_{ii}\ne 0$.

Write a proof