+coercion cic:/matita/dama/ordered_uniform/segment_square_of_ordered_set_square.con 0 2.
+
+alias symbol "pi1" (instance 4) = "exT \fst".
+alias symbol "pi1" (instance 2) = "exT \fst".
+lemma ordered_set_square_of_segment_square :
+ ∀O:ordered_set.∀u,v:O.{[u,v]} square → O square ≝
+ λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion cic:/matita/dama/ordered_uniform/ordered_set_square_of_segment_square.con.
+
+lemma restriction_agreement :
+ ∀O:ordered_uniform_space.∀l,r:O.∀P:{[l,r]} square → Prop.∀OP:O square → Prop.Prop.
+apply(λO:ordered_uniform_space.λl,r:O.
+ λP:{[l,r]} square → Prop.λOP:O square → Prop.
+ ∀b:O square.∀H1,H2.(P b → OP b) ∧ (OP b → P b));
+[5,7: apply H1|6,8:apply H2]skip;
+qed.
+
+lemma unrestrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} square.
+ restriction_agreement ? l r U u → U x → u x.
+intros 7; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b b1 x;
+cases (H 〈w1,w2〉 H1 H2) (L _); intro Uw; apply L; apply Uw;
+qed.
+
+lemma restrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} square.
+ restriction_agreement ? l r U u → u x → U x.
+intros 6; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b1 b x;
+intros (Ra uw); cases (Ra 〈w1,w2〉 H1 H2) (_ R); apply R; apply uw;
+qed.
+
+lemma invert_restriction_agreement:
+ ∀O:ordered_uniform_space.∀l,r:O.
+ ∀U:{[l,r]} square → Prop.∀u:O square → Prop.
+ restriction_agreement ? l r U u →
+ restriction_agreement ? l r (inv U) (inv u).
+intros 9; split; intro;
+[1: apply (unrestrict ????? (segment_square_of_ordered_set_square ??? 〈\snd b,\fst b〉 H2 H1) H H3);
+|2: apply (restrict ????? (segment_square_of_ordered_set_square ??? 〈\snd b,\fst b〉 H2 H1) H H3);]
+qed.
+
+alias symbol "square" (instance 8) = "bishop set square".
+lemma bs_of_ss:
+ ∀O:ordered_set.∀u,v:O.{[u,v]} square → (bishop_set_of_ordered_set O) square ≝
+ λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈\fst(\fst b),\fst(\snd b)〉.
+
+notation < "x \sub \neq" with precedence 91 for @{'bsss $x}.
+interpretation "bs_of_ss" 'bsss x = (bs_of_ss _ _ _ x).
+
+alias symbol "square" (instance 7) = "ordered set square".
+lemma ss_of_bs:
+ ∀O:ordered_set.∀u,v:O.
+ ∀b:O square.\fst b ∈ [u,v] → \snd b ∈ [u,v] → {[u,v]} square ≝
+ λO:ordered_set.λu,v:O.
+ λb:(O:bishop_set) square.λH1,H2.〈≪\fst b,H1≫,≪\snd b,H2≫〉.
+
+notation < "x \sub \nleq" with precedence 91 for @{'ssbs $x}.
+interpretation "ss_of_bs" 'ssbs x = (ss_of_bs _ _ _ x _ _).