reserve I for non empty set;
reserve M for ManySortedSet of I;
reserve Y,x,y,y1,i,j for set;
reserve k for Element of NAT;
reserve p for FinSequence;
reserve S for non void non empty ManySortedSign;
reserve A for non-empty MSAlgebra over S;

theorem Th5:
  EqRelLatt Y is complete
proof
  for X being Subset of EqRelLatt Y ex a being Element of EqRelLatt Y st a
is_less_than X & for b being Element of EqRelLatt Y st b is_less_than X holds b
  [= a
  proof
    let X be Subset of EqRelLatt Y;
    per cases;
    suppose
A1:   X is empty;
      take a = Top EqRelLatt Y;
      for q be Element of EqRelLatt Y st q in X holds a [= q by A1;
      hence a is_less_than X by LATTICE3:def 16;
      let b be Element of EqRelLatt Y;
      assume b is_less_than X;
      thus thesis by LATTICES:19;
    end;
    suppose
A2:   X is non empty;
      set a = meet X;
      X c= bool [:Y,Y:]
      proof
        let x be object;
        assume x in X;
        then x is Equivalence_Relation of Y by MSUALG_5:21;
        hence thesis;
      end;
      then reconsider X9 = X as Subset-Family of [:Y,Y:];
      for Z be set st Z in X holds Z is Equivalence_Relation of Y by
MSUALG_5:21;
      then meet X9 is Equivalence_Relation of Y by A2,EQREL_1:11;
      then reconsider a as Equivalence_Relation of Y;
      set a9 = a;
      reconsider a as Element of EqRelLatt Y by MSUALG_5:21;
      take a;
      now
        let q be Element of EqRelLatt Y;
        reconsider q9 = q as Equivalence_Relation of Y by MSUALG_5:21;
        assume q in X;
        then a9 c= q9 by SETFAM_1:3;
        hence a [= q by Th2;
      end;
      hence a is_less_than X by LATTICE3:def 16;
      now
        let b be Element of EqRelLatt Y;
        reconsider b9 = b as Equivalence_Relation of Y by MSUALG_5:21;
        assume
A3:     b is_less_than X;
        now
          let x be object;
          assume
A4:       x in b9;
          now
            let Z be set;
            assume
A5:         Z in X;
            then reconsider Z1 = Z as Element of EqRelLatt Y;
            reconsider Z9 = Z1 as Equivalence_Relation of Y by MSUALG_5:21;
            b [= Z1 by A3,A5,LATTICE3:def 16;
            then b9 c= Z9 by Th2;
            hence x in Z by A4;
          end;
          hence x in meet X by A2,SETFAM_1:def 1;
        end;
        then b9 c= meet X;
        hence b [= a by Th2;
      end;
      hence thesis;
    end;
  end;
  hence thesis by VECTSP_8:def 6;
end;
