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 Th7:
  TS is T_2 & PS is compact implies PS is closed
proof
  assume that
A1: TS is T_2 and
A2: PS is compact;
  per cases;
  suppose
    PS = {};
    hence thesis;
  end;
  suppose
A3: PS <> {};
    now
      let a be set;
      thus a in PS` implies ex Q being Subset of TS st Q is open & Q c= PS` &
      a in Q
      proof
        assume
A4:     a in PS`;
        then reconsider p=a as Point of TS;
        consider W,V being Subset of TS such that
A5:     W is open and
        V is open and
A6:     p in W and
A7:     PS c= V and
A8:     W misses V by A1,A2,A3,A4,Th6;
        take Q = W;
        W misses V`` by A8;
        then
A9:     W c= V` by SUBSET_1:24;
        V` c= PS` by A7,SUBSET_1:12;
        hence thesis by A5,A6,A9;
      end;
      thus (ex Q being Subset of TS st Q is open & Q c= PS` & a in Q) implies
      a in PS`;
    end;
    then PS` is open by TOPS_1:25;
    hence thesis;
  end;
end;
