reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;
reserve S for non empty TopStruct;
reserve f for Function of T,S;
reserve SS for non empty TopSpace;
reserve f for Function of TS,SS;

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;
