"to get a contradiction"?Your approach was based on combining everything to get a contradiction. How does that work if you don't combine rectangles?
You don't necessarily have to write it like that, depends on the direction you take. Doesn't matter."to get a contradiction"?
Well, you claim that, but can you prove it? It doesn't work as you described in my example. It will work if I make a complete tiling out of it, but that is not the point.I claim that there is always an order of rectangles which satisfies the third lemma( both vertically and horizontally), and if not, there is the forth lemma.