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 E1 be Enumeration of F1 holds
    doms (SignGenOp(f,A,F1) * E1) c= doms (SignGenOp(f^g,A,F1) * E1)
proof
  let E1 be Enumeration of F1;
  let x such that
A1: x in doms (SignGenOp(f,A,F1) * E1);
  reconsider x as FinSequence by A1;
A2: len (SignGenOp(f,A,F1) * E1)=len E1 by CARD_1:def 7;
  then
A3: len x = len E1 by A1,Th47;
  then
A4: len x = len (SignGenOp(f^g,A,F1) * E1) by CARD_1:def 7;
  then
A5: dom x = dom (SignGenOp(f^g,A,F1) * E1) by FINSEQ_3:30;
A6: dom x = dom (SignGenOp(f,A,F1) * E1) by A2,A3,FINSEQ_3:30;
  for i st i in dom x holds x.i in dom ((SignGenOp(f^g,A,F1) * E1).i)
  proof
    let i such that
A7:   i in dom x;
A8:   dom f c= dom (f^g) by FINSEQ_1:26;
A9:   x.i in dom ((SignGenOp(f,A,F1) * E1).i) by A7,A1,Th47;
    (SignGenOp(f,A,F1) * E1).i = SignGen(f,A,E1.i) by Th80,A7,A6;
    then len((SignGenOp(f,A,F1) * E1).i) = len (f) by CARD_1:def 7;
    then
A10:  dom((SignGenOp(f,A,F1) * E1).i) = dom (f) by FINSEQ_3:30;
    (SignGenOp(f^g,A,F1) * E1).i = SignGen(f^g,A,E1.i) by A7,A5,Th80;
    then len((SignGenOp(f^g,A,F1) * E1).i) = len (f^g) by CARD_1:def 7;
    then dom((SignGenOp(f^g,A,F1) * E1).i) = dom (f^g) by FINSEQ_3:30;
    hence thesis by A8,A9,A10;
  end;
  hence thesis by A4,Th47;
end;
