set f = ComSign A;
defpred S₁[ Nat, set ] means $2 = [[:(ProdOp (A,$1)):]];
A1: for k being Nat st k in Seg (len (ComSign A)) holds
ex x being Element of PFuncs (((product (Carrier A)) *),(product (Carrier A))) st S₁[k,x]

proof

let k be Nat; :: thesis: ( k in Seg (len (ComSign A)) implies ex x being Element of PFuncs (((product (Carrier A)) *),(product (Carrier A))) st S₁[k,x] )
assume k in Seg (len (ComSign A)) ; :: thesis: ex x being Element of PFuncs (((product (Carrier A)) *),(product (Carrier A))) st S₁[k,x]
reconsider a = [[:(ProdOp (A,k)):]] as Element of PFuncs (((product (Carrier A)) *),(product (Carrier A))) by PARTFUN1:45;
take a ; :: thesis: S₁[k,a]
thus S₁[k,a] ; :: thesis: verum

end;

consider p being FinSequence of PFuncs (((product (Carrier A)) *),(product (Carrier A))) such that
A2: ( dom p = Seg (len (ComSign A)) & ( for k being Nat st k in Seg (len (ComSign A)) holds
S₁[k,p . k] ) ) from FINSEQ_1:sch 5(A1);
reconsider p = p as PFuncFinSequence of (product (Carrier A)) ;
take p ; :: thesis: ( len p = len (ComSign A) & ( for n being Nat st n in dom p holds
p . n = [[:(ProdOp (A,n)):]] ) )
thus len p = len (ComSign A) by A2, FINSEQ_1:def 3; :: thesis: for n being Nat st n in dom p holds
p . n = [[:(ProdOp (A,n)):]]
let n be Nat; :: thesis: ( n in dom p implies p . n = [[:(ProdOp (A,n)):]] )
assume n in dom p ; :: thesis: p . n = [[:(ProdOp (A,n)):]]
hence p . n = [[:(ProdOp (A,n)):]] by A2; :: thesis: verum