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 Th102:
  for E1 be Enumeration of F1 st n in dom f holds
    (len E1)|-> n in doms(SignGenOp(f,A,F1) * E1)
proof
  let E1 be Enumeration of F1 such that
A1: n in dom f;
  set CE=SignGenOp(f,A,F1) * E1, s=(len E1)|-> n;
A2: len  CE=len E1 =len s by CARD_1:def 7;
  then
A3: dom s = dom E1= dom CE by FINSEQ_3:29;
  for i st i in dom s holds s.i in dom (CE.i)
  proof
    let i such that
A4:   i in dom s;
    s.i = n & dom f = dom SignGen(f,A,E1.i) by Def11,A4,FINSEQ_2:57;
    hence thesis by A1,A3,A4,Th80;
  end;
  hence thesis by Th47,A2;
end;
