 reserve U for set,
         X, Y for Subset of U;
 reserve U for non empty set,
         A, B, C for non empty IntervalSet of U;

theorem
  A _\/_ (B _/\_ C) = (A _\/_ B) _/\_ (A _\/_ C)
  proof
A1: A _\/_ (B _/\_ C) c= (A _\/_ B) _/\_ (A _\/_ C) by Lm1;
    (A _\/_ B) _/\_ (A _\/_ C) c= A _\/_ (B _/\_ C)
    proof
      let x be object;
      assume x in (A _\/_ B) _/\_ (A _\/_ C); then
      consider X, Y being set such that
A2:   X in UNION (A,B) & Y in UNION (A,C) & x = X /\ Y by SETFAM_1:def 5;
A3:   A is non empty ordered Subset-Family of U &
        B is non empty ordered Subset-Family of U &
        C is non empty ordered Subset-Family of U by Lm4;
      x in INTERSECTION (UNION (A,B), UNION (A,C)) by A2,SETFAM_1:def 5;
      hence thesis by Th29,A3;
    end;
    hence thesis by A1;
  end;
