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 Th19:
  X0 is SubSpace of X1 & (X1 misses X2 or X2 misses X1) implies X0
  misses X2 & X2 misses 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 X0 is SubSpace of X1;
  then
A1: A0 c= A1 by TSEP_1:4;
A2: now
    assume X1 misses X2;
    then A2 misses A1 by TSEP_1:def 3;
    hence thesis by TSEP_1:def 3,A1,XBOOLE_1:63;
  end;
  assume X1 misses X2 or X2 misses X1;
  hence thesis by A2;
end;
