reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X1, X2, X3 for non empty SubSpace of X;
reserve X for TopSpace;
reserve A1, A2 for Subset of X;
reserve A1,A2 for Subset of X;
reserve X for TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace,
  A1, A2 for Subset of X;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;

theorem
  X1,X2 are_separated iff ex Y1, Y2 being closed non empty SubSpace of X
  st X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & (Y1 misses Y2 or Y1 meet Y2
  misses X1 union X2)
proof
  reconsider A2 = the carrier of X2 as Subset of X by Th1;
  reconsider A1 = the carrier of X1 as Subset of X by Th1;
  thus X1,X2 are_separated implies ex Y1, Y2 being closed non empty SubSpace
of X st X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & (Y1 misses Y2 or Y1 meet
  Y2 misses X1 union X2)
  proof
    assume X1,X2 are_separated;
    then A1,A2 are_separated;
    then consider C1, C2 being Subset of X such that
A1: A1 c= C1 and
A2: A2 c= C2 and
A3: C1 /\ C2 misses A1 \/ A2 and
A4: C1 is closed and
A5: C2 is closed by Th43;
    C1 is non empty by A1,XBOOLE_1:3;
    then consider Y1 being strict closed non empty SubSpace of X such that
A6: C1 = the carrier of Y1 by A4,Th15;
    C2 is non empty by A2,XBOOLE_1:3;
    then consider Y2 being strict closed non empty SubSpace of X such that
A7: C2 = the carrier of Y2 by A5,Th15;
    take Y1,Y2;
    now
      assume not Y1 misses Y2;
      then
A8:   the carrier of Y1 meet Y2 = C1 /\ C2 by A6,A7,Def4;
      the carrier of X1 union X2 = A1 \/ A2 by Def2;
      hence Y1 meet Y2 misses X1 union X2 by A3,A8;
    end;
    hence thesis by A1,A2,A6,A7,Th4;
  end;
  given Y1, Y2 being closed non empty SubSpace of X such that
A9: X1 is SubSpace of Y1 & X2 is SubSpace of Y2 and
A10: Y1 misses Y2 or Y1 meet Y2 misses X1 union X2;
  now
    let A1, A2 be Subset of X such that
A11: A1 = the carrier of X1 & A2 = the carrier of X2;
    ex C1 being Subset of X, C2 being Subset of X st A1 c= C1 & A2 c= C2
    & C1 /\ C2 misses A1 \/ A2 & C1 is closed & C2 is closed
    proof
      reconsider C2 = the carrier of Y2 as Subset of X by Th1;
      reconsider C1 = the carrier of Y1 as Subset of X by Th1;
      take C1,C2;
      now
        per cases;
        suppose
          Y1 misses Y2;
          then C1 misses C2;
          then C1 /\ C2 = {} by XBOOLE_0:def 7;
          hence C1 /\ C2 misses A1 \/ A2 by XBOOLE_1:65;
        end;
        suppose
A12:      not Y1 misses Y2;
A13:      the carrier of X1 union X2 = A1 \/ A2 by A11,Def2;
          the carrier of Y1 meet Y2 = C1 /\ C2 by A12,Def4;
          hence C1 /\ C2 misses A1 \/ A2 by A10,A12,A13;
        end;
      end;
      hence thesis by A9,A11,Th4,Th11;
    end;
    hence A1,A2 are_separated by Th43;
  end;
  hence thesis;
end;
