let p, q be Element of HP-WFF ; :: thesis: ( p is canonical & p => q is canonical implies q is canonical )
assume that
A1: p is canonical and
A2: p => q is canonical ; :: thesis: q 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,q)
consider x being set such that
A3: for P being Permutation of V holds x is_a_fixpoint_of Perm (P,p) by A1;
set P = the Permutation of V;
A4: dom (Perm ( the Permutation of V,(p => q))) = SetVal (V,(p => q)) by FUNCT_2:52
.= Funcs ((SetVal (V,p)),(SetVal (V,q))) by Def2 ;
consider f being set such that
A5: for P being Permutation of V holds f is_a_fixpoint_of Perm (P,(p => q)) by A2;
f is_a_fixpoint_of Perm ( the Permutation of V,(p => q)) by A5;
then reconsider f = f as Function of (SetVal (V,p)),(SetVal (V,q)) by FUNCT_2:66, A4;
take f . x ; :: thesis: for P being Permutation of V holds f . x is_a_fixpoint_of Perm (P,q)
let P be Permutation of V; :: thesis: f . x is_a_fixpoint_of Perm (P,q)
A6: f is_a_fixpoint_of Perm (P,(p => q)) by A5;
x is_a_fixpoint_of Perm (P,p) by A3;
hence f . x is_a_fixpoint_of Perm (P,q) by A6, Th39; :: thesis: verum