+nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
+ #A; napply mk_binary_morphism1
+ [ napply (λS,S': qpowerclass_setoid ?. S ⊆ S')
+ | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
+ [ napply (subseteq_trans … a' a) (* anche qui, perche' serve a'? *)
+ [ nassumption | napply (subseteq_trans … a b); nassumption ]
+ ##| napply (subseteq_trans … a a') (* anche qui, perche' serve a'? *)
+ [ nassumption | napply (subseteq_trans … a' b'); nassumption ] ##]
+nqed.
+
+nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
+ #A; napply mk_binary_morphism1
+ [ #S; #S'; napply mk_qpowerclass
+ [ napply (S ∩ S')
+ | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); napply mk_iff; *; #H1; #H2; napply conj
+ [##1,2: napply (. (mem_ok' …)^-1) [##3,6: nassumption |##1,4: nassumption |##*: ##skip]
+ ##|##3,4: napply (. (mem_ok' …)) [##2,5: nassumption |##1,4: nassumption |##*: ##skip]##]##]
+ ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; napply conj; #x; nwhd in ⊢ (% → %); #H
+ [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+ | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
+nqed.
+
+unification hint 0 (∀A.∀U,V.(λx,y.True) (fun21 ??? (intersect_ok A) U V) (intersect A U V)).
+
+nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
+ #A; #U; #V; #x; #x'; #H; #p;
+ (* CSC: senza la change non funziona! *)
+ nchange with (x' ∈ (fun21 ??? (intersect_ok A) U V));
+ napply (. (H^-1‡#)); nassumption.
+nqed.