let F, G be Element of Funcs ([:NAT,(Funcs (bound_QC-variables,bound_QC-variables)):],CQC-WFF); :: thesis: ( ( for k being Element of NAT
for h being Element of Funcs (bound_QC-variables,bound_QC-variables)
for p being Element of CQC-WFF st p = f . ((k + 1),(h +* (x .--> (x. k)))) holds
F . (k,h) = All ((x. k),p) ) & ( for k being Element of NAT
for h being Element of Funcs (bound_QC-variables,bound_QC-variables)
for p being Element of CQC-WFF st p = f . ((k + 1),(h +* (x .--> (x. k)))) holds
G . (k,h) = All ((x. k),p) ) implies F = G )
assume A3: for k being Element of NAT
for h being Element of Funcs (bound_QC-variables,bound_QC-variables)
for p being Element of CQC-WFF st p = f . ((k + 1),(h +* (x .--> (x. k)))) holds
F . (k,h) = All ((x. k),p) ; :: thesis: ( ex k being Element of NAT ex h being Element of Funcs (bound_QC-variables,bound_QC-variables) ex p being Element of CQC-WFF st
( p = f . ((k + 1),(h +* (x .--> (x. k)))) & not G . (k,h) = All ((x. k),p) ) or F = G )
assume A4: for k being Element of NAT
for h being Element of Funcs (bound_QC-variables,bound_QC-variables)
for p being Element of CQC-WFF st p = f . ((k + 1),(h +* (x .--> (x. k)))) holds
G . (k,h) = All ((x. k),p) ; :: thesis: F = G
for a being Element of [:NAT,(Funcs (bound_QC-variables,bound_QC-variables)):] holds F . a = G . a

proof

let a be Element of [:NAT,(Funcs (bound_QC-variables,bound_QC-variables)):]; :: thesis: F . a = G . a
consider k being Element of NAT , h being Element of Funcs (bound_QC-variables,bound_QC-variables) such that
A5: a = [k,h] by DOMAIN_1:1;
reconsider h2 = h +* (x .--> (x. k)) as Function of bound_QC-variables,bound_QC-variables by Lm1;
reconsider h2 = h2 as Element of Funcs (bound_QC-variables,bound_QC-variables) by FUNCT_2:8;
reconsider p = f . ((k + 1),h2) as Element of CQC-WFF ;
F . (k,h) = All ((x. k),p) by A3
.= G . (k,h) by A4 ;
hence F . a = G . a by A5; :: thesis: verum

end;

hence F = G by FUNCT_2:63; :: thesis: verum