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 Th28:
  X1 is SubSpace of X0 & (X0 misses X2 or X2 misses X0) implies X0
  meet (X1 union X2) = the TopStruct of X1 & X0 meet (X2 union X1) = the
  TopStruct of 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;
A1: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  assume
A2: X1 is SubSpace of X0;
  then
A3: A1 c= A0 by TSEP_1:4;
  assume X0 misses X2 or X2 misses X0;
  then
A4: A0 misses A2 by TSEP_1:def 3;
  X0 meets X1 by A2,Th17;
  then X0 meets X1 union X2 by A1,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
    .= A1 by A3,XBOOLE_1:28,A4;
  hence X0 meet (X1 union X2) = the TopStruct of X1 by TSEP_1:5;
  thus thesis by A5,TSEP_1:5;
end;
