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 Th110:
  for E be Enumeration of F, p be Permutation of dom E
    for s be FinSequence st s in doms (SignGenOp(f,A,F) * E)
      holds s*p in doms (SignGenOp(f,A,F) * (E * p))
proof
  let E be Enumeration of F,
      p be Permutation of dom E;
  let s be FinSequence such that
A1: s in doms (SignGenOp(f,A,F) * E);
  reconsider Ep=E*p as Enumeration of F by Th109;
A2: len (SignGenOp(f,A,F) * Ep) = len Ep = card F by CARD_1:def 7;
A3: len s = len (SignGenOp(f,A,F) * E) = len E =card F by Th47,A1,CARD_1:def 7;
  then
A4: dom E = dom s by FINSEQ_3:30;
  dom p = dom E = rng p by FUNCT_2:52,def 3;
  then
A5: dom (s*p) = dom s = Seg len s by A4,RELAT_1:27,FINSEQ_1:def 3;
  reconsider sp=s*p as FinSequence by A4;
A6: len sp = len s by A5,FINSEQ_3:29;
  for i st i in dom sp holds sp.i in dom ((SignGenOp(f,A,F) * Ep).i)
  proof
    let i;
    assume i in dom sp;
    then
A7:   i in dom p & p.i in dom s by FUNCT_1:11;
    then
A8:   s.(p.i) in dom ((SignGenOp(f,A,F) * E).(p.i)) & s.(p.i)=sp.i
      by A1,Th47,FUNCT_1:13;
    ((SignGenOp(f,A,F) * E).(p.i)) = ((SignGenOp(f,A,F) * E)*p).i
      by A7,FUNCT_1:13
    .= (SignGenOp(f,A,F) * Ep).i by RELAT_1:36;
    hence thesis by A8;
  end;
  hence thesis by A2,A3,A6,Th47;
end;
