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 Th70:
  X misses dom f implies SignGen(f,B,X) = f
proof
  assume
A1: X misses dom f;
A2: dom f = dom SignGen(f,B,X) by Def11;
  for i st i in dom f holds SignGen(f,B,X).i = f.i
  proof
    let i such that
A3:   i in dom f;
    not i in X by A3,A1,XBOOLE_0:3;
    hence thesis by A2,A3,Def11;
  end;
  hence thesis by A2;
end;
