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;
