defpred S₁[ Element of HP-WFF ] means Perm (P,V) is bijective ;
A1: for n being Element of NAT holds S₁[ prop n]

proof

let n be Element of NAT ; :: thesis: S₁[ prop n]
( SetVal (V,(prop n)) = V . n & Perm (P,(prop n)) = P . n ) by Def2, Def5;
hence S₁[ prop n] by Def4; :: thesis: verum

end;

A2: for r, s being Element of HP-WFF st S₁[r] & S₁[s] holds
( S₁[r '&' s] & S₁[r => s] )

proof

let r, s be Element of HP-WFF ; :: thesis: ( S₁[r] & S₁[s] implies ( S₁[r '&' s] & S₁[r => s] ) )
assume Perm (P,r) is bijective ; :: thesis: ( not S₁[s] or ( S₁[r '&' s] & S₁[r => s] ) )
then reconsider r9 = Perm (P,r) as Permutation of (SetVal (V,r)) ;
assume Perm (P,s) is bijective ; :: thesis: ( S₁[r '&' s] & S₁[r => s] )
then reconsider s9 = Perm (P,s) as Permutation of (SetVal (V,s)) ;
( SetVal (V,(r '&' s)) = [:(SetVal (V,r)),(SetVal (V,s)):] & Perm (P,(r '&' s)) = [:r9,s9:] ) by Def2, Th34;
hence Perm (P,(r '&' s)) is bijective ; :: thesis: S₁[r => s]
( SetVal (V,(r => s)) = Funcs ((SetVal (V,r)),(SetVal (V,s))) & Perm (P,(r => s)) = r9 => s9 ) by Def2, Th35;
hence S₁[r => s] ; :: thesis: verum

end;

Perm (P,VERUM) = id (SetVal (V,VERUM)) by Th32;
then A3: S₁[ VERUM ] ;
for p being Element of HP-WFF holds S₁[p] from HILBERT2:sch 2(A3, A1, A2);
hence Perm (P,p) is bijective ; :: thesis: verum