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;

theorem Th9:
  PS is compact & QS c= PS & QS is closed implies QS is compact
proof
  assume that
A1: PS is compact and
A2: QS c= PS and
A3: QS is closed;
  per cases;
  suppose
    PS = {};
    hence thesis by A2;
  end;
  suppose
    PS <> {};
    then TS|PS is compact by A1,Th3;
    then
A4: for QQ being Subset of TS|PS st QQ=QS holds QQ is compact by A3,Th8,
TOPS_2:26;
    PS = [#] (TS|PS) by PRE_TOPC:def 5;
    hence thesis by A2,A4,Th2;
  end;
end;
