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 Th106:
  for E be Enumeration of F holds
    doms(SignGenOp(f,A,F)*E) = doms(len f,card F)
proof
  let E be Enumeration of F;
A1: len (SignGenOp(f,A,F)*E) = len E = card F by CARD_1:def 7;
  thus doms (SignGenOp(f,A,F)*E) c= doms(len f,card F)
  proof
    let x such that
A2:   x in doms (SignGenOp(f,A,F)*E);
    reconsider x as FinSequence by A2;
A3:   len x = len (SignGenOp(f,A,F)*E) by A2,Th47;
    rng x c= Seg len f
    proof
      let y;
      assume y in rng x;
      then consider i be object such that A4:  i in dom x & y=x.i
        by FUNCT_1:def 3;
      reconsider i as Nat by A4;
A5:     x.i in dom ((SignGenOp(f,A,F)*E).i) by A4,A2,Th47;
      (SignGenOp(f,A,F)*E).i<>{} by A4,A2,Th47;
      then i in dom (SignGenOp(f,A,F)*E) by FUNCT_1:def 2;
      then (SignGenOp(f,A,F) * E).i = SignGen(f,A,E.i) by Th80;
      then x.i in dom f by A5,Def11;
      hence thesis by A4,FINSEQ_1:def 3;
    end;
    then reconsider x as FinSequence of Seg len f by FINSEQ_1:def 4;
    x in (Seg len f)* by FINSEQ_1:def 11;
    hence thesis by A1,A3;
  end;
  let x such that
A6: x in doms(len f,card F);
  consider s be Element of (Seg len f)* such that
A7: x=s & len s= card F by A6;
A8: rng s c= Seg len f =dom f by FINSEQ_1:def 3;
  for i st i in dom s holds s.i in dom ((SignGenOp(f,A,F)*E).i)
  proof
    let i such that
A9:  i in dom s;
A10:  s.i in rng s by A9,FUNCT_1:def 3;
    dom s = dom (SignGenOp(f,A,F)*E) by A1,A7,FINSEQ_3:29;
    then (SignGenOp(f,A,F) * E).i = SignGen(f,A,E.i) by A9,Th80;
    then dom ((SignGenOp(f,A,F)*E).i) = dom f by Def11;
    hence thesis by A8,A10;
  end;
  hence thesis by A7,A1,Th47;
end;
