(* Constructions with subset_equivalence ************************************)
+lemma subset_equivalence_ext_f1_exteq (A1) (A0) (f1) (f2) (u):
+ f1 ⊜ f2 → subset_ext_f1 A1 A0 f1 u ⇔ subset_ext_f1 A1 A0 f2 u.
+/3 width=3 by subset_inclusion_ext_f1_exteq, conj/
+qed.
+
lemma subset_equivalence_ext_f1_bi (A1) (A0) (f) (u1) (v1):
u1 ⇔ v1 → subset_ext_f1 A1 A0 f u1 ⇔ subset_ext_f1 A1 A0 f v1.
#A1 #A0 #f #u1 #v1 * #Huv1 #Hvu1
/3 width=3 by subset_inclusion_ext_f1_bi, conj/
qed.
+
+lemma subset_inclusion_ext_f1_compose (A0) (A1) (A2) (f1) (f2) (u):
+ subset_ext_f1 A1 A2 f2 (subset_ext_f1 A0 A1 f1 u) ⇔ subset_ext_f1 A0 A2 (f2∘f1) u.
+/3 width=1 by subset_inclusion_ext_f1_compose_dx, subset_inclusion_ext_f1_compose_sn, conj/
+qed.