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 Th30:
  X1 is SubSpace of Y1 & X2 is SubSpace of Y2 & (Y1 misses Y2 or
  Y1 meet Y2 misses X1 union X2) implies Y1 misses X2 & Y2 misses X1
proof
  assume that
A1: X1 is SubSpace of Y1 and
A2: X2 is SubSpace of Y2;
  assume
A3: Y1 misses Y2 or Y1 meet Y2 misses X1 union X2;
  now
    assume
A4: not Y1 misses Y2;
A5: now
      assume Y2 meets X1;
      then
A6:   (Y1 meet Y2) meets X1 by A1,Th29;
      X1 is SubSpace of X1 union X2 by TSEP_1:22;
      hence contradiction by A3,A4,A6,Th18;
    end;
    now
      assume Y1 meets X2;
      then
A7:   (Y1 meet Y2) meets X2 by A2,Th29;
      X2 is SubSpace of X1 union X2 by TSEP_1:22;
      hence contradiction by A3,A4,A7,Th18;
    end;
    hence thesis by A5;
  end;
  hence thesis by A1,A2,Th19;
end;
