I also came up with a way to avoid what I think is using the theorem itself, but it doesn't involve the latter four inequalities at all. For the first one I did: $x^R+y\leq x+z$, so none of $x^R+y\geq (x+z)^R$, and thus $x^R+y \ngeq x^R+z$, but from $y\geq z$ and induction follows $x^R+y \geq x^R+z$, so a contradiction. The third one is similar. The second one implies by $y^R\leq z$ using the first part of the proof, but $y\geq z$ implies $y^R\nleq z$, so a contradiction. The fourth one is similar.

I think I understand the structure of the proof, but to get from the first four inequalities to the latter four, I can't see an easy way without using the theorem itself. I do understand the rest though.

I edited the question to make it more readable and clear what I don't understand