reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem
  for E be Enumeration of F, p be Permutation of dom E
  for S be Element of Fin doms(n,card F) holds
   {s*p where s is FinSequence of NAT: s in S}
     is Element of Fin doms(n,card F)
proof
  let E be Enumeration of F, p be Permutation of dom E;
  reconsider Ep = E*p as Enumeration of F by Th109;
  set D = the non empty set;
  set f = the n-element FinSequence of D;
  set A = the BinOp of D;
A1: len f = n by CARD_1:def 7;
  let S be Element of Fin doms(n,card F);
    {s*p where s is FinSequence of NAT: s in S} c= doms(n,card F)
  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
A2:   y=s*p & s in S;
    S c= doms(n,card F) by FINSUB_1:def 5;
    then S c= dom App (SignGenOp(f,A,F) * E) by A1,Lm3;
    then s in dom App (SignGenOp(f,A,F) * E) by A2;
    then s*p in doms (SignGenOp(f,A,F) * Ep) by Th110;
    hence thesis by A2,A1,Th106;
  end;
  hence thesis by FINSUB_1:def 5;
end;
