+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.