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 Th18:
  X0 is SubSpace of X1 & (X0 meets X2 or X2 meets X0) implies X1
  meets X2 & X2 meets 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;
  assume X0 is SubSpace of X1;
  then
A1: A0 c= A1 by TSEP_1:4;
A2: now
    assume X0 meets X2;
    then A2 meets A0 by TSEP_1:def 3;
    hence thesis by TSEP_1:def 3,A1,XBOOLE_1:63;
  end;
  assume X0 meets X2 or X2 meets X0;
  hence thesis by A2;
end;
