I think the proof will be something like this, since the leftmost non zeri entry in first r rows didnt change it so property 2 still holds as the entries are still 1 and also propert 1 of RREF still holds as there are r non zero rows then r+1 to m zero rows property 3 also holds as the columns of the left most entries disnt change and property 4 holds as the column and rows of leftmost non zero entries disnt change and since it was true before adding column j it is true after addinb the column,

Is this a good proof

I was proof the bit in second paragraph