reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,i for set;
reserve r,r1,r2 for Real;

theorem
  Bottom EqRelLatt M = id M
proof
  set K = id M;
  K is Equivalence_Relation of M by Th1;
  then reconsider K as Element of EqRelLatt M by MSUALG_5:def 5;
  now
    let a be Element of EqRelLatt M;
    reconsider K9 = K, a9 = a as Equivalence_Relation of M by MSUALG_5:def 5;
A1: now
      let i be object;
      assume
A2:   i in I;
      then reconsider i9 = i as Element of I;
      reconsider a2 = a9.i9 as Equivalence_Relation of M.i by MSUALG_4:def 2;
      thus (K9 (/\) a9).i = K9.i /\ a9.i by A2,PBOOLE:def 5
        .= id (M.i) /\ a2 by MSUALG_3:def 1
        .= id (M.i) by EQREL_1:10
        .= K9.i by A2,MSUALG_3:def 1;
    end;
    thus K "/\" a = (the L_meet of EqRelLatt M).(K,a) by LATTICES:def 2
      .= K9 (/\) a9 by MSUALG_5:def 5
      .= K by A1,PBOOLE:3;
    hence a "/\" K = K;
  end;
  hence thesis by LATTICES:def 16;
end;
