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 Th27:
  X1 meets X2 implies (X1 is SubSpace of X0 implies X1 meet X2 is
SubSpace of X0 meet X2) & (X2 is SubSpace of X0 implies X1 meet X2 is SubSpace
  of X1 meet X0)
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;
  assume
A1: X1 meets X2;
  then
A2: the carrier of X1 meet X2 = A1 /\ A2 by TSEP_1:def 4;
A3: now
    assume
A4: X2 is SubSpace of X0;
    then
A5: A1 /\ A2 c= A1 /\ A0 by XBOOLE_1:26,TSEP_1:4;
    X1 meets X0 by A1,A4,Th18;
    then the carrier of X1 meet X0 = A1 /\ A0 by TSEP_1:def 4;
    hence X1 meet X2 is SubSpace of X1 meet X0 by A2,A5,TSEP_1:4;
  end;
  now
    assume
A6: X1 is SubSpace of X0;
    then A1 c= A0 by TSEP_1:4;
    then
A7: A1 /\ A2 c= A0 /\ A2 by XBOOLE_1:26;
    X0 meets X2 by A1,A6,Th18;
    then the carrier of X0 meet X2 = A0 /\ A2 by TSEP_1:def 4;
    hence X1 meet X2 is SubSpace of X0 meet X2 by A2,A7,TSEP_1:4;
  end;
  hence thesis by A3;
end;
