theorem
  for f be PartFunc of CNS,CNS st f = id dom f holds f is_continuous_on dom f
proof
  let f be PartFunc of CNS,CNS;
  assume
A1: f = id dom f;
  now
    let x0 be Point of CNS such that
A2: x0 in dom f;
    thus f/.x0=f.x0 by A2,PARTFUN1:def 6
      .= x0 by A1,A2,FUNCT_1:17;
  end;
  hence thesis by Th128;
end;
