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 Th26:
  ((X1 misses X0 or X0 misses X1) & (X2 meets X0 or X0 meets X2)
implies (X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2
) & ((X1 meets X0 or X0 meets X1) & (X2 misses X0 or X0 misses X2) implies (X1
  union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1)
proof
  reconsider A0 = the carrier of X0, A1 = the carrier of X1, A2 = the carrier
  of X2 as Subset of X by TSEP_1:1;
  thus (X1 misses X0 or X0 misses X1) & (X2 meets X0 or X0 meets X2) implies (
  X1 union X2) meet X0 = X2 meet X0 & X0 meet (X1 union X2) = X0 meet X2
  proof
    assume that
A1: X1 misses X0 or X0 misses X1 and
A2: X2 meets X0 or X0 meets X2;
A3: A1 misses A0 by A1,TSEP_1:def 3;
    X2 is SubSpace of X1 union X2 by TSEP_1:22;
    then
A4: (X1 union X2) meets X0 by A2,Th18;
    then
A5: the carrier of X0 meet (X1 union X2) = A0 /\ (the carrier of (X1 union
    X2)) by TSEP_1:def 4
      .= A0 /\ (A1 \/ A2) by TSEP_1:def 2
      .= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23
      .= the carrier of (X0 meet X2) by A2,TSEP_1:def 4,A3;
    the carrier of (X1 union X2) meet X0 = (the carrier of (X1 union X2))
    /\ A0 by A4,TSEP_1:def 4
      .= (A1 \/ A2) /\ A0 by TSEP_1:def 2
      .= (A1 /\ A0) \/ (A2 /\ A0) by XBOOLE_1:23
      .= the carrier of (X2 meet X0) by A2,TSEP_1:def 4,A3;
    hence thesis by A5,TSEP_1:5;
  end;
  thus (X1 meets X0 or X0 meets X1) & (X2 misses X0 or X0 misses X2) implies (
  X1 union X2) meet X0 = X1 meet X0 & X0 meet (X1 union X2) = X0 meet X1
  proof
    assume that
A6: X1 meets X0 or X0 meets X1 and
A7: X2 misses X0 or X0 misses X2;
A8: A2 misses A0 by A7,TSEP_1:def 3;
    X1 is SubSpace of X1 union X2 by TSEP_1:22;
    then
A9: (X1 union X2) meets X0 by A6,Th18;
    then
A10: the carrier of X0 meet (X1 union X2) = A0 /\ (the carrier of (X1
    union X2)) by TSEP_1:def 4
      .= A0 /\ (A1 \/ A2) by TSEP_1:def 2
      .= (A0 /\ A1) \/ (A0 /\ A2) by XBOOLE_1:23
      .= the carrier of (X0 meet X1) by A6,TSEP_1:def 4,A8;
    the carrier of (X1 union X2) meet X0 = (the carrier of (X1 union X2))
    /\ A0 by A9,TSEP_1:def 4
      .= (A1 \/ A2) /\ A0 by TSEP_1:def 2
      .= (A1 /\ A0) \/ {} by A8,XBOOLE_1:23
      .= the carrier of (X1 meet X0) by A6,TSEP_1:def 4;
    hence thesis by A10,TSEP_1:5;
  end;
end;
