theorem Th8:
  c in dom (f^) implies f/.c <> 0
proof
  assume that
A1: c in dom (f^) and
A2: (f/.c) = 0;
A3: c in dom f \ f"{0c} by A1,Def2;
  then
A4: not c in f"{0c} by XBOOLE_0:def 5;
  now
    per cases by A4,PARTFUN2:26;
    suppose
      not c in dom f;
      hence contradiction by A3,XBOOLE_0:def 5;
    end;
    suppose
      not (f/.c) in {0c};
      hence contradiction by A2,TARSKI:def 1;
    end;
  end;
  hence contradiction;
end;
