theorem Th25:
  the L_meet of L is_distributive_wrt the L_meet of L
proof
  now
    let a,b,c;
    thus (the L_meet of L).(a,(the L_meet of L).(b,c)) = a "/\" (b "/\" c)
      .= a "/\" b "/\" c by LATTICES:def 7
      .= a "/\" a "/\" b "/\" c
      .= (a "/\" b) "/\" a "/\" c by LATTICES:def 7
      .= (a "/\" b) "/\" (a "/\" c) by LATTICES:def 7
      .= (the L_meet of L).((the L_meet of L).(a,b),(the L_meet of L).(a,c));
  end;
  hence thesis by BINOP_1:12;
end;
