reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;

theorem Th33:
  X1 meets X2 & X1 is not SubSpace of X2 & X2 is not SubSpace of
  X1 & not X1 union X2 is SubSpace of Y1 union Y2 & the TopStruct of X = (Y1
  union Y2) union X0 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1
  union X2) is SubSpace of X2 & X0 meet (X1 union X2) is SubSpace of X1 meet X2
  implies Y1 union Y2 meets X1 union X2 & X0 meets X1 union X2
proof
  assume
A1: X1 meets X2;
  assume
A2: X1 is not SubSpace of X2 & X2 is not SubSpace of X1;
  reconsider C = the carrier of X0 as Subset of X by TSEP_1:1;
  reconsider C2 = the carrier of Y2 as Subset of X by TSEP_1:1;
  reconsider C1 = the carrier of Y1 as Subset of X by TSEP_1:1;
  reconsider A2 = the carrier of X2 as Subset of X by TSEP_1:1;
  reconsider A1 = the carrier of X1 as Subset of X by TSEP_1:1;
A3: the carrier of Y1 union Y2 = C1 \/ C2 by TSEP_1:def 2;
A4: Y1 is SubSpace of Y1 union Y2 by TSEP_1:22;
  assume
A5: not X1 union X2 is SubSpace of Y1 union Y2;
  assume
A6: the TopStruct of X = (Y1 union Y2) union X0;
  assume
A7: Y1 meet (X1 union X2) is SubSpace of X1 & Y2 meet (X1 union X2) is
  SubSpace of X2;
  assume X0 meet (X1 union X2) is SubSpace of X1 meet X2;
  then Y1 meets X1 union X2 by A1,A2,A6,A7,Th32;
  hence Y1 union Y2 meets X1 union X2 by A4,Th18;
A8: the carrier of X1 union X2 = A1 \/ A2 by TSEP_1:def 2;
  then
A9: not A1 \/ A2 c= C1 \/ C2 by A5,A3,TSEP_1:4;
  now
    assume X0 misses (X1 union X2);
    then
A10: C misses (A1 \/ A2) by A8,TSEP_1:def 3;
    the carrier of X = (C1 \/ C2) \/ C by A6,A3,TSEP_1:def 2;
    then A1 \/ A2 = ((C1 \/ C2) \/ C) /\ (A1 \/ A2) by XBOOLE_1:28
      .= ((C1 \/ C2) /\ (A1 \/ A2)) \/ (C /\ (A1 \/ A2)) by XBOOLE_1:23
      .= (C1 \/ C2) /\ (A1 \/ A2) by A10;
    hence contradiction by A9,XBOOLE_1:17;
  end;
  hence thesis;
end;
