lemma square_ordered_set: ordered_set → ordered_set.
intro O;
apply (mk_ordered_set (O × O));
-[1: intros (x y); apply (fst x ≰ fst y ∨ snd x ≰ snd y);
+[1: intros (x y); apply (\fst x ≰ \fst y ∨ \snd x ≰ \snd y);
|2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
cases H (X X); apply (os_coreflexive ?? X);
|3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2);