
theorem Th3:
  for X, Y being Subset of REAL holds Cl(X \/ Y) = Cl X \/ Cl Y
proof
  let X, Y be Subset of REAL;
  reconsider A = X, B = Y as Subset of R^1 by TOPMETR:17;
  thus Cl(X \/ Y) = Cl(A \/ B) by JORDAN5A:24
    .= Cl A \/ Cl B by PRE_TOPC:20
    .= Cl X \/ Cl B by JORDAN5A:24
    .= Cl X \/ Cl Y by JORDAN5A:24;
end;
