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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/formal_topology/".
+include "logic/equality.ma".
+
+axiom S: Type.
+
+axiom leq: S → S → Prop.
+
+notation "hvbox(A break ⊆ B)" with precedence 59
+for @{ 'subseteq $A $B}.
+
+interpretation "Subseteq" 'subseteq A B =
+ (cic:/matita/formal_topology/leq.con A B).
+
+axiom leq_refl: ∀A. A ⊆ A.
+axiom leq_antisym: ∀A,B. A ⊆ B → B ⊆ A → A=B.
+axiom leq_tran: ∀A,B,C. A ⊆ B → B ⊆ C → A ⊆ C.
+
+axiom i: S → S.
+
+axiom i_contrattivita: ∀A. i A ⊆ A.
+axiom i_idempotenza: ∀A. i (i A) = i A.
+axiom i_monotonia: ∀A,B. A ⊆ B → i A ⊆ i B.
+
+axiom c: S → S.
+
+axiom c_espansivita: ∀A. A ⊆ c A.
+axiom c_idempotenza: ∀A. c (c A) = c A.
+axiom c_monotonia: ∀A,B. A ⊆ B → c A ⊆ c B.
+
+axiom m: S → S.
+
+axiom m_antimonotonia: ∀A,B. A ⊆ B → m B ⊆ m A.
+axiom m_saturazione: ∀A. A ⊆ m (m A).
+axiom m_puntofisso: ∀A. m A = m (m (m A)).
+
+lemma l1: ∀A,B. i A ⊆ B → i A ⊆ i B.
+ intros; rewrite < i_idempotenza; apply (i_monotonia (i A) B H).
+qed.
+lemma l2: ∀A,B. A ⊆ c B → c A ⊆ c B.
+ intros; rewrite < c_idempotenza in ⊢ (? ? %); apply (c_monotonia A (c B) H).
+qed.
+
+axiom th1: ∀A. c (m A) ⊆ m (i A).
+axiom th2: ∀A. i (m A) ⊆ m (c A).
+
+(************** start of generated part *********************)
+