Prove that you can multiply using these steps

Question

How to prove this?

1. Think of a simple multiplication problem, I will use $22 \times 7$

2. Divide the first factor by 2; if the number isn't a whole number, floor it (e. g. $11 / 2 = 5$), until you come to $1$

3. Multiply the second factor by $2$ with increasing power ($7 \times 2^0, 7 \times 2^1$ etc.) until you make as many numbers as you made in step 1, make "columns".

4. Remove every column that has an even number in its upper cell.

5. Add all number in the second line together.

This works universally for whole numbers.

e. g.

$22;11;5;2;1$

$7;14;28;56;112$

this becomes

$11;5;1$

$14;28;112$

$22 \times 7 = 14 + 28 + 112 = 154$

Thanks in advance!