reserve Y for non empty TopStruct;

theorem Th8:
  for A, B being Subset of Y st A is closed or B is closed holds A
  is T_0 & B is T_0 implies A \/ B is T_0
proof
  let A, B be Subset of Y;
  assume
A1: A is closed or B is closed;
  assume that
A2: A is T_0 and
A3: B is T_0;
  now
    let x, y be Point of Y;
    assume
A4: x in A \/ B & y in A \/ B;
    then
A5: x in (A \ B) \/ B & y in (A \ B) \/ B by XBOOLE_1:39;
    assume
A6: x <> y;
A7: x in A \/ (B \ A) & y in A \/ (B \ A) by A4,XBOOLE_1:39;
    now
      per cases by A1;
      suppose
A8:     A is closed;
        now
          per cases by A7,XBOOLE_0:def 3;
          suppose
            x in A & y in A;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
or ex W being Subset of Y st W is closed & not x in W & y in W by A2,A6;
          end;
          suppose
A9:         x in A & y in B \ A;
            now
              take A;
              thus A is closed by A8;
              thus x in A by A9;
              thus not y in A by A9,XBOOLE_0:def 5;
            end;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
            or ex W being Subset of Y st W is closed & not x in W & y in W;
          end;
          suppose
A10:        x in B \ A & y in A;
            now
              take A;
              thus A is closed by A8;
              thus not x in A by A10,XBOOLE_0:def 5;
              thus y in A by A10;
            end;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
            or ex W being Subset of Y st W is closed & not x in W & y in W;
          end;
          suppose
A11:        x in B \ A & y in B \ A;
            B \ A c= B by XBOOLE_1:36;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
or ex W being Subset of Y st W is closed & not x in W & y in W by A3,A6,A11;
          end;
        end;
        hence (ex V being Subset of Y st V is closed & x in V & not y in V) or
        ex W being Subset of Y st W is closed & not x in W & y in W;
      end;
      suppose
A12:    B is closed;
        now
          per cases by A5,XBOOLE_0:def 3;
          suppose
A13:        x in A \ B & y in A \ B;
            A \ B c= A by XBOOLE_1:36;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
or ex W being Subset of Y st W is closed & not x in W & y in W by A2,A6,A13;
          end;
          suppose
A14:        x in A \ B & y in B;
            now
              take B;
              thus B is closed by A12;
              thus not x in B by A14,XBOOLE_0:def 5;
              thus y in B by A14;
            end;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
            or ex W being Subset of Y st W is closed & not x in W & y in W;
          end;
          suppose
A15:        x in B & y in A \ B;
            now
              take B;
              thus B is closed by A12;
              thus x in B by A15;
              thus not y in B by A15,XBOOLE_0:def 5;
            end;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
            or ex W being Subset of Y st W is closed & not x in W & y in W;
          end;
          suppose
            x in B & y in B;
            hence
            (ex V being Subset of Y st V is closed & x in V & not y in V)
or ex W being Subset of Y st W is closed & not x in W & y in W by A3,A6;
          end;
        end;
        hence (ex V being Subset of Y st V is closed & x in V & not y in V) or
        ex W being Subset of Y st W is closed & not x in W & y in W;
      end;
    end;
    hence
    (ex V being Subset of Y st V is closed & x in V & not y in V) or ex W
    being Subset of Y st W is closed & not x in W & y in W;
  end;
  hence thesis;
end;
