theorem Th113:
  for E be Enumeration of F, p be Permutation of dom E
  for S be Element of Fin dom App (SignGenOp(f,A,F) * E)
     holds
   {s*p where s is FinSequence of NAT: s in S}
     is Element of Fin dom App (SignGenOp(f,A,F) * (E*p))
proof
  let E be Enumeration of F, p be Permutation of dom E;
  let S be Element of Fin dom App (SignGenOp(f,A,F) * E);
  {s*p where s is FinSequence of NAT: s in S} c=
    dom App (SignGenOp(f,A,F) * (E*p))
  proof
    let y;
    assume y in {s*p where s is FinSequence of NAT: s in S};
    then consider s be FinSequence of NAT such that
A1:   y=s*p & s in S;
    S c= dom App (SignGenOp(f,A,F) * E) by FINSUB_1:def 5;
    then s in dom App (SignGenOp(f,A,F) * E) by A1;
    then s*p in doms (SignGenOp(f,A,F) * (E*p)) by Th110;
    hence thesis by A1,Def9;
  end;
  hence thesis by FINSUB_1:def 5;
end;
