reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2, X3 for non empty SubSpace of X;

theorem
  (X1 union X2) union X3 = X1 union (X2 union X3)
proof
  the carrier of (X1 union X2) union X3 = (the carrier of X1 union X2) \/
  (the carrier of X3) by Def2
    .= ((the carrier of X1) \/ (the carrier of X2)) \/ (the carrier of X3)
  by Def2
    .= (the carrier of X1) \/ ((the carrier of X2) \/ (the carrier of X3))
  by XBOOLE_1:4
    .= (the carrier of X1) \/ (the carrier of X2 union X3) by Def2
    .= the carrier of X1 union (X2 union X3) by Def2;
  hence thesis by Th5;
end;
