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 Th31:
  X1 is not SubSpace of X2 & X2 is not SubSpace of X1 & X1 union
  X2 is SubSpace of Y1 union Y2 & Y1 meet (X1 union X2) is SubSpace of X1 & Y2
meet (X1 union X2) is SubSpace of X2 implies Y1 meets X1 union X2 & Y2 meets X1
  union X2
proof
  assume that
A1: X1 is not SubSpace of X2 and
A2: X2 is not SubSpace of X1;
  reconsider A1 = the carrier of X1, A2 = the carrier of X2, C1 = the carrier
  of Y1, C2 = the carrier of Y2 as Subset of X by TSEP_1:1;
  assume
A3: X1 union X2 is SubSpace of Y1 union Y2;
  assume that
A4: Y1 meet (X1 union X2) is SubSpace of X1 and
A5: Y2 meet (X1 union X2) is SubSpace of X2;
A6: the carrier of X1 union X2 = A1 \/ A2 by TSEP_1:def 2;
A7: the carrier of Y1 union Y2 = C1 \/ C2 by TSEP_1:def 2;
A8: now
    assume Y2 misses (X1 union X2);
    then
A9: C2 misses (A1 \/ A2) by A6,TSEP_1:def 3;
    A1 \/ A2 c= C1 \/ C2 by A3,A6,A7,TSEP_1:4;
    then
A10: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
      .= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
      .= C1 /\ (A1 \/ A2) by A9;
    then C1 meets (A1 \/ A2);
    then Y1 meets (X1 union X2) by A6,TSEP_1:def 3;
    then the carrier of Y1 meet (X1 union X2) = C1 /\ (A1 \/ A2) by A6,
TSEP_1:def 4;
    then
A11: A1 \/ A2 c= A1 by A4,A10,TSEP_1:4;
    A2 c= A1 \/ A2 by XBOOLE_1:7;
    hence contradiction by A2,TSEP_1:4,A11,XBOOLE_1:1;
  end;
  now
    assume Y1 misses (X1 union X2);
    then
A12: C1 misses (A1 \/ A2) by A6,TSEP_1:def 3;
    A1 \/ A2 c= C1 \/ C2 by A3,A6,A7,TSEP_1:4;
    then
A13: A1 \/ A2 = (C1 \/ C2) /\ (A1 \/ A2) by XBOOLE_1:28
      .= (C1 /\ (A1 \/ A2)) \/ (C2 /\ (A1 \/ A2)) by XBOOLE_1:23
      .= C2 /\ (A1 \/ A2) by A12;
    then C2 meets (A1 \/ A2);
    then Y2 meets (X1 union X2) by A6,TSEP_1:def 3;
    then the carrier of Y2 meet (X1 union X2) = C2 /\ (A1 \/ A2) by A6,
TSEP_1:def 4;
    then
A14: A1 \/ A2 c= A2 by A5,A13,TSEP_1:4;
    A1 c= A1 \/ A2 by XBOOLE_1:7;
    then A1 c= A2 by A14,XBOOLE_1:1;
    hence contradiction by A1,TSEP_1:4;
  end;
  hence thesis by A8;
end;
