reserve I,X,x,d,i for set;
reserve M for ManySortedSet of I;
reserve EqR1,EqR2 for Equivalence_Relation of X;
reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve EqR,EqR1,EqR2,EqR3,EqR4 for Equivalence_Relation of M;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem
  Bottom CongrLatt A = id (the Sorts of A)
proof
  set K = id (the Sorts of A);
  K is MSCongruence of A by Th17;
  then reconsider K as Element of CongrLatt A by Def6;
A1: the L_meet of CongrLatt A = (the L_meet of EqRelLatt the Sorts of A)||
  the carrier of CongrLatt A by NAT_LAT:def 12;
  now
    let a be Element of CongrLatt A;
    reconsider K9 = K, a9 = a as Equivalence_Relation of the Sorts of A by Def6
;
A2: [K,a] in [:the carrier of CongrLatt A,the carrier of CongrLatt A:] by
ZFMISC_1:87;
A3: now
      let i be object;
      assume
A4:   i in the carrier of S;
      then reconsider i9 = i as Element of S;
      reconsider a2 = a9.i9 as Equivalence_Relation of (the Sorts of A).i by
MSUALG_4:def 2;
      thus (K9 (/\) a9).i = K9.i /\ a9.i by A4,PBOOLE:def 5
        .= id ((the Sorts of A).i) /\ a2 by MSUALG_3:def 1
        .= id ((the Sorts of A).i) by EQREL_1:10
        .= K9.i by A4,MSUALG_3:def 1;
    end;
    thus K "/\" a = (the L_meet of CongrLatt A).(K,a) by LATTICES:def 2
      .= (the L_meet of EqRelLatt the Sorts of A).(K,a) by A1,A2,FUNCT_1:49
      .= K9 (/\) a9 by Def5
      .= K by A3,PBOOLE:3;
    hence a "/\" K = K;
  end;
  hence thesis by LATTICES:def 16;
end;
