theorem Th20:
  the L_join of L is_distributive_wrt the L_join of L
proof
  now
    let a,b,c;
    thus (the L_join of L).(a,(the L_join of L).(b,c)) = a "\/" (b "\/" c)
      .= a "\/" b "\/" c by LATTICES:def 5
      .= a "\/" a "\/" b "\/" c
      .= (a "\/" b) "\/" a "\/" c by LATTICES:def 5
      .= (a "\/" b) "\/" (a "\/" c) by LATTICES:def 5
      .= (the L_join of L).((the L_join of L).(a,b),(the L_join of L).(a,c));
  end;
  hence thesis by BINOP_1:12;
end;
