-[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 (hos_coreflexive ?? X);
-|3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2);
- clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H;
- [1: cases (hos_cotransitive ??? z1 H1); [left; left|right;left]assumption;
- |2: cases (hos_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+[1: apply (wloss O);
+|2: intros; cases (wloss_prop O); [left|right] intros; apply H;
+|3: apply (square_exc O);
+|4: intro x; cases (wloss_prop O); rewrite < (H ?? (square_exc O) x x); clear H;
+ cases x; clear x; unfold square_exc; intro H; cases H; clear H; simplify in H1;
+ [1,3: apply (hos_coreflexive O h H1);
+ |*: apply (hos_coreflexive O h1 H1);]
+|5: intros 3 (x0 y0 z0); cases (wloss_prop O);
+ do 3 rewrite < (H ?? (square_exc O)); clear H; cases x0; cases y0; cases z0; clear x0 y0 z0;
+ simplify; intro H; cases H; clear H;
+ [1: cases (hos_cotransitive ? h h2 h4 H1); [left;left|right;left] assumption;
+ |2: cases (hos_cotransitive ? h1 h3 h5 H1); [left;right|right;right] assumption;
+ |3: cases (hos_cotransitive ? h2 h h4 H1); [right;left|left;left] assumption;
+ |4: cases (hos_cotransitive ? h3 h1 h5 H1); [right;right|left;right] assumption;]]