+lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A ∩ B =1 C ∩ B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+
+lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A ∩ B =1 A ∩ C.
+#U #A #B #C #eqBC #a @iff_and_l @eqBC qed.
+
+lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A - B =1 C - B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+
+lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A - B =1 A - C.
+#U #A #B #C #eqBC #a @iff_and_l /2/ qed.
+