defpred S₁[ Element of NAT , set , set ] means for p, q being Element of CQC-WFF st p = f . [$1,$2] & q = g . [($1 + n),$2] holds
$3 = p '&' q;
A1: for k being Element of NAT
for h being Element of Funcs bound_QC-variables ,bound_QC-variables ex u being Element of CQC-WFF st S₁[k,h,u]

proof

let k be Element of NAT ; :: thesis: for h being Element of Funcs bound_QC-variables ,bound_QC-variables ex u being Element of CQC-WFF st S₁[k,h,u]
let h be Element of Funcs bound_QC-variables ,bound_QC-variables ; :: thesis: ex u being Element of CQC-WFF st S₁[k,h,u]
reconsider p = f . [k,h] as Element of CQC-WFF ;
reconsider q = g . [(k + n),h] as Element of CQC-WFF ;
take p '&' q ; :: thesis: S₁[k,h,p '&' q]
let p1, q1 be Element of CQC-WFF ; :: thesis: ( p1 = f . [k,h] & q1 = g . [(k + n),h] implies p '&' q = p1 '&' q1 )
assume that
A2: p1 = f . [k,h] and
A3: q1 = g . [(k + n),h] ; :: thesis: p '&' q = p1 '&' q1
thus p '&' q = p1 '&' q1 by A2, A3; :: thesis: verum

end;

consider F being Function of [:NAT ,(Funcs bound_QC-variables ,bound_QC-variables ):],CQC-WFF such that
A4: for k being Element of NAT
for h being Element of Funcs bound_QC-variables ,bound_QC-variables holds S₁[k,h,F . k,h] from BINOP_1:sch 3(A1);
reconsider F = F as Element of Funcs [:NAT ,(Funcs bound_QC-variables ,bound_QC-variables ):],CQC-WFF by FUNCT_2:11;
take F ; :: thesis: for k being Element of NAT
for h being Element of Funcs bound_QC-variables ,bound_QC-variables
for p, q being Element of CQC-WFF st p = f . k,h & q = g . (k + n),h holds
F . k,h = p '&' q
let k be Element of NAT ; :: thesis: for h being Element of Funcs bound_QC-variables ,bound_QC-variables
for p, q being Element of CQC-WFF st p = f . k,h & q = g . (k + n),h holds
F . k,h = p '&' q
let h be Element of Funcs bound_QC-variables ,bound_QC-variables ; :: thesis: for p, q being Element of CQC-WFF st p = f . k,h & q = g . (k + n),h holds
F . k,h = p '&' q
let p, q be Element of CQC-WFF ; :: thesis: ( p = f . k,h & q = g . (k + n),h implies F . k,h = p '&' q )
assume that
A5: p = f . k,h and
A6: q = g . (k + n),h ; :: thesis: F . k,h = p '&' q
thus F . k,h = p '&' q by A4, A5, A6; :: thesis: verum