theorem Th16:
  TS is compact & SS is T_2 & rng f = [#] SS & f is continuous
  implies for PS st PS is closed holds f.:PS is closed
proof
  assume that
A1: TS is compact and
A2: SS is T_2 and
A3: rng f = [#] SS and
A4: f is continuous;
  let PS;
  assume PS is closed;
  then PS is compact by A1,Th8;
  hence thesis by A2,A3,A4,Th7,Th15;
end;
