let p, q be Element of HP-WFF ; :: thesis: (p '&' q) => p is canonical
let V be SetValuation; :: according to HILBERT3:def 7 :: thesis: ex x being set st
for P being Permutation of V holds x is_a_fixpoint_of Perm (P,((p '&' q) => p))
take f = pr1 ((SetVal (V,p)),(SetVal (V,q))); :: thesis: for P being Permutation of V holds f is_a_fixpoint_of Perm (P,((p '&' q) => p))
let P be Permutation of V; :: thesis: f is_a_fixpoint_of Perm (P,((p '&' q) => p))
A1: dom (Perm (P,((p '&' q) => p))) = SetVal (V,((p '&' q) => p)) by FUNCT_2:def 1
.= Funcs ((SetVal (V,(p '&' q))),(SetVal (V,p))) by Def2
.= Funcs ([:(SetVal (V,p)),(SetVal (V,q)):],(SetVal (V,p))) by Def2 ;
hence f in dom (Perm (P,((p '&' q) => p))) by FUNCT_2:11; :: according to ABIAN:def 3 :: thesis: f = (Perm (P,((p '&' q) => p))) . f
then f in Funcs ((SetVal (V,(p '&' q))),(SetVal (V,p))) by A1, Def2;
then reconsider F = f as Function of (SetVal (V,(p '&' q))),(SetVal (V,p)) by FUNCT_2:121;
thus (Perm (P,((p '&' q) => p))) . f = ((Perm (P,p)) * F) * ((Perm (P,(p '&' q))) ") by Th37
.= ((Perm (P,p)) * F) * ([:(Perm (P,p)),(Perm (P,q)):] ") by Th35
.= ((Perm (P,p)) * F) * [:((Perm (P,p)) "),((Perm (P,q)) "):] by FUNCTOR0:7
.= (Perm (P,p)) * (F * [:((Perm (P,p)) "),((Perm (P,q)) "):]) by RELAT_1:55
.= (Perm (P,p)) * (((Perm (P,p)) ") * F) by Th15
.= ((Perm (P,p)) * ((Perm (P,p)) ")) * F by RELAT_1:55
.= (id (SetVal (V,p))) * F by FUNCT_2:88
.= f by FUNCT_2:23 ; :: thesis: verum