Article LATTICE8, MML version 4.99.1005

:: LATTICE8:attrnot 1 => LATTICE8:attr 1
definition
  let a1 be RelStr;
  attr a1 is finitely_typed means
    ex b1 being non empty set st
       (for b2 being set
              st b2 in the carrier of a1
           holds b2 is total symmetric transitive Relation of b1,b1) &
        (ex b2 being Element of NAT st
           for b3, b4 being total symmetric transitive Relation of b1,b1
           for b5, b6 being set
                 st b3 in the carrier of a1 & b4 in the carrier of a1 & [b5,b6] in b3 "\/" b4
              holds ex b7 being non empty FinSequence of b1 st
                 len b7 = b2 & b5,b6 are_joint_by b7,b3,b4);
end;

:: LATTICE8:dfs 1
definiens
  let a1 be RelStr;
To prove
     a1 is finitely_typed
it is sufficient to prove
  thus ex b1 being non empty set st
       (for b2 being set
              st b2 in the carrier of a1
           holds b2 is total symmetric transitive Relation of b1,b1) &
        (ex b2 being Element of NAT st
           for b3, b4 being total symmetric transitive Relation of b1,b1
           for b5, b6 being set
                 st b3 in the carrier of a1 & b4 in the carrier of a1 & [b5,b6] in b3 "\/" b4
              holds ex b7 being non empty FinSequence of b1 st
                 len b7 = b2 & b5,b6 are_joint_by b7,b3,b4);

:: LATTICE8:def 2
theorem
for b1 being RelStr holds
      b1 is finitely_typed
   iff
      ex b2 being non empty set st
         (for b3 being set
                st b3 in the carrier of b1
             holds b3 is total symmetric transitive Relation of b2,b2) &
          (ex b3 being Element of NAT st
             for b4, b5 being total symmetric transitive Relation of b2,b2
             for b6, b7 being set
                   st b4 in the carrier of b1 & b5 in the carrier of b1 & [b6,b7] in b4 "\/" b5
                holds ex b8 being non empty FinSequence of b2 st
                   len b8 = b3 & b6,b7 are_joint_by b8,b4,b5);

:: LATTICE8:prednot 1 => LATTICE8:pred 1
definition
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a2 be Element of NAT;
  pred A1 has_a_representation_of_type<= A2 means
    ex b1 being non trivial set st
       ex b2 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of EqRelLATT b1 st
          b2 is one-to-one &
           Image b2 is finitely_typed &
           (ex b3 being total symmetric transitive Relation of b1,b1 st
              b3 in the carrier of Image b2 & b3 <> id b1) &
           type_of Image b2 <= a2;
end;

:: LATTICE8:dfs 2
definiens
  let a1 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a2 be Element of NAT;
To prove
     a1 has_a_representation_of_type<= a2
it is sufficient to prove
  thus ex b1 being non trivial set st
       ex b2 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of a1,the carrier of EqRelLATT b1 st
          b2 is one-to-one &
           Image b2 is finitely_typed &
           (ex b3 being total symmetric transitive Relation of b1,b1 st
              b3 in the carrier of Image b2 & b3 <> id b1) &
           type_of Image b2 <= a2;

:: LATTICE8:def 3
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2 being Element of NAT holds
      b1 has_a_representation_of_type<= b2
   iff
      ex b3 being non trivial set st
         ex b4 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b3 st
            b4 is one-to-one &
             Image b4 is finitely_typed &
             (ex b5 being total symmetric transitive Relation of b3,b3 st
                b5 in the carrier of Image b4 & b5 <> id b3) &
             type_of Image b4 <= b2;

:: LATTICE8:exreg 1
registration
  cluster non empty finite reflexive transitive antisymmetric lower-bounded distributive with_suprema with_infima RelStr;
end;

:: LATTICE8:exreg 2
registration
  let a1 be non trivial set;
  cluster non empty non trivial full meet-inheriting join-inheriting finitely_typed SubRelStr of EqRelLATT a1;
end;

:: LATTICE8:th 1
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2 holds
   succ {} c= DistEsti b3;

:: LATTICE8:th 2
theorem
for b1 being trivial reflexive transitive antisymmetric with_infima RelStr holds
   b1 is modular;

:: LATTICE8:th 3
theorem
for b1 being non empty set
for b2 being non empty meet-inheriting join-inheriting SubRelStr of EqRelLATT b1
      st b2 is not trivial
   holds ex b3 being total symmetric transitive Relation of b1,b1 st
      b3 in the carrier of b2 & b3 <> id b1;

:: LATTICE8:th 4
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
      st b3 is infs-preserving(b1, b2) & b3 is sups-preserving(b1, b2)
   holds b3 is meet-preserving(b1, b2) & b3 is join-preserving(b1, b2);

:: LATTICE8:th 5
theorem
for b1, b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
      st b1,b2 are_isomorphic & b1 is modular
   holds b2 is modular;

:: LATTICE8:th 6
theorem
for b1 being non empty reflexive transitive antisymmetric lower-bounded RelStr
for b2 being non empty reflexive transitive antisymmetric RelStr
for b3 being Function-like quasi_total monotone Relation of the carrier of b1,the carrier of b2 holds
   Image b3 is lower-bounded;

:: LATTICE8:th 7
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b2, b3 being Element of the carrier of b1
for b4 being non empty set
for b5 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b4
      st b5 is one-to-one &
         (corestr b5) . b2 <= (corestr b5) . b3
   holds b2 <= b3;

:: LATTICE8:th 8
theorem
for b1 being non trivial set
for b2 being non empty full meet-inheriting join-inheriting finitely_typed SubRelStr of EqRelLATT b1
for b3 being total symmetric transitive Relation of b1,b1
      st b3 in the carrier of b2 & b3 <> id b1 & type_of b2 <= 2
   holds b2 is modular;

:: LATTICE8:th 9
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
      st b1 has_a_representation_of_type<= 2
   holds b1 is modular;

:: LATTICE8:funcnot 1 => LATTICE8:func 1
definition
  let a1 be set;
  func new_set2 A1 -> set equals
    a1 \/ {{a1},{{a1}}};
end;

:: LATTICE8:def 4
theorem
for b1 being set holds
   new_set2 b1 = b1 \/ {{b1},{{b1}}};

:: LATTICE8:funcreg 1
registration
  let a1 be set;
  cluster new_set2 a1 -> non empty;
end;

:: LATTICE8:funcnot 2 => LATTICE8:func 2
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  let a4 be Element of [:a1,a1,the carrier of a2,the carrier of a2:];
  func new_bi_fun2(A3,A4) -> Function-like quasi_total Relation of [:new_set2 a1,new_set2 a1:],the carrier of a2 means
    (for b1, b2 being Element of a1 holds
     it .(b1,b2) = a3 .(b1,b2)) &
     it .({a1},{a1}) = Bottom a2 &
     it .({{a1}},{{a1}}) = Bottom a2 &
     it .({a1},{{a1}}) = ((a3 .(a4 `1,a4 `2)) "\/" (a4 `3)) "/\" (a4 `4) &
     it .({{a1}},{a1}) = ((a3 .(a4 `1,a4 `2)) "\/" (a4 `3)) "/\" (a4 `4) &
     (for b1 being Element of a1 holds
        it .(b1,{a1}) = (a3 .(b1,a4 `1)) "\/" (a4 `3) &
         it .({a1},b1) = (a3 .(b1,a4 `1)) "\/" (a4 `3) &
         it .(b1,{{a1}}) = (a3 .(b1,a4 `2)) "\/" (a4 `3) &
         it .({{a1}},b1) = (a3 .(b1,a4 `2)) "\/" (a4 `3));
end;

:: LATTICE8:def 5
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:]
for b5 being Function-like quasi_total Relation of [:new_set2 b1,new_set2 b1:],the carrier of b2 holds
      b5 = new_bi_fun2(b3,b4)
   iff
      (for b6, b7 being Element of b1 holds
       b5 .(b6,b7) = b3 .(b6,b7)) &
       b5 .({b1},{b1}) = Bottom b2 &
       b5 .({{b1}},{{b1}}) = Bottom b2 &
       b5 .({b1},{{b1}}) = ((b3 .(b4 `1,b4 `2)) "\/" (b4 `3)) "/\" (b4 `4) &
       b5 .({{b1}},{b1}) = ((b3 .(b4 `1,b4 `2)) "\/" (b4 `3)) "/\" (b4 `4) &
       (for b6 being Element of b1 holds
          b5 .(b6,{b1}) = (b3 .(b6,b4 `1)) "\/" (b4 `3) &
           b5 .({b1},b6) = (b3 .(b6,b4 `1)) "\/" (b4 `3) &
           b5 .(b6,{{b1}}) = (b3 .(b6,b4 `2)) "\/" (b4 `3) &
           b5 .({{b1}},b6) = (b3 .(b6,b4 `2)) "\/" (b4 `3));

:: LATTICE8:th 10
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
   st b3 is zeroed(b1, b2)
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:] holds
   new_bi_fun2(b3,b4) is zeroed(new_set2 b1, b2);

:: LATTICE8:th 11
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
   st b3 is symmetric(b1, b2)
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:] holds
   new_bi_fun2(b3,b4) is symmetric(new_set2 b1, b2);

:: LATTICE8:th 12
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
   st b2 is modular
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
   st b3 is symmetric(b1, b2) & b3 is u.t.i.(b1, b2)
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:]
      st b3 .(b4 `1,b4 `2) <= b4 `3 "\/" (b4 `4)
   holds new_bi_fun2(b3,b4) is u.t.i.(new_set2 b1, b2);

:: LATTICE8:th 14
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being Element of [:b1,b1,the carrier of b2,the carrier of b2:] holds
   b3 c= new_bi_fun2(b3,b4);

:: LATTICE8:funcnot 3 => LATTICE8:func 3
definition
  let a1 be non empty set;
  let a2 be ordinal set;
  func ConsecutiveSet2(A1,A2) -> set means
    ex b1 being Relation-like Function-like T-Sequence-like set st
       it = last b1 &
        proj1 b1 = succ a2 &
        b1 . {} = a1 &
        (for b2 being ordinal set
              st succ b2 in succ a2
           holds b1 . succ b2 = new_set2 (b1 . b2)) &
        (for b2 being ordinal set
              st b2 in succ a2 & b2 <> {} & b2 is being_limit_ordinal
           holds b1 . b2 = union proj2 (b1 | b2));
end;

:: LATTICE8:def 6
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being set holds
      b3 = ConsecutiveSet2(b1,b2)
   iff
      ex b4 being Relation-like Function-like T-Sequence-like set st
         b3 = last b4 &
          proj1 b4 = succ b2 &
          b4 . {} = b1 &
          (for b5 being ordinal set
                st succ b5 in succ b2
             holds b4 . succ b5 = new_set2 (b4 . b5)) &
          (for b5 being ordinal set
                st b5 in succ b2 & b5 <> {} & b5 is being_limit_ordinal
             holds b4 . b5 = union proj2 (b4 | b5));

:: LATTICE8:th 15
theorem
for b1 being non empty set holds
   ConsecutiveSet2(b1,{}) = b1;

:: LATTICE8:th 16
theorem
for b1 being non empty set
for b2 being ordinal set holds
   ConsecutiveSet2(b1,succ b2) = new_set2 ConsecutiveSet2(b1,b2);

:: LATTICE8:th 17
theorem
for b1 being non empty set
for b2 being ordinal set
for b3 being Relation-like Function-like T-Sequence-like set
      st b2 <> {} &
         b2 is being_limit_ordinal &
         proj1 b3 = b2 &
         (for b4 being ordinal set
               st b4 in b2
            holds b3 . b4 = ConsecutiveSet2(b1,b4))
   holds ConsecutiveSet2(b1,b2) = union proj2 b3;

:: LATTICE8:funcreg 2
registration
  let a1 be non empty set;
  let a2 be ordinal set;
  cluster ConsecutiveSet2(a1,a2) -> non empty;
end;

:: LATTICE8:th 18
theorem
for b1 being non empty set
for b2 being ordinal set holds
   b1 c= ConsecutiveSet2(b1,b2);

:: LATTICE8:funcnot 4 => LATTICE8:func 4
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  let a4 be QuadrSeq of a3;
  let a5 be ordinal set;
  assume a5 in proj1 a4;
  func Quadr2(A4,A5) -> Element of [:ConsecutiveSet2(a1,a5),ConsecutiveSet2(a1,a5),the carrier of a2,the carrier of a2:] equals
    a4 . a5;
end;

:: LATTICE8:def 7
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set
      st b5 in proj1 b4
   holds Quadr2(b4,b5) = b4 . b5;

:: LATTICE8:funcnot 5 => LATTICE8:func 5
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  let a4 be QuadrSeq of a3;
  let a5 be ordinal set;
  func ConsecutiveDelta2(A4,A5) -> set means
    ex b1 being Relation-like Function-like T-Sequence-like set st
       it = last b1 &
        proj1 b1 = succ a5 &
        b1 . {} = a3 &
        (for b2 being ordinal set
              st succ b2 in succ a5
           holds b1 . succ b2 = new_bi_fun2(BiFun(b1 . b2,ConsecutiveSet2(a1,b2),a2),Quadr2(a4,b2))) &
        (for b2 being ordinal set
              st b2 in succ a5 & b2 <> {} & b2 is being_limit_ordinal
           holds b1 . b2 = union proj2 (b1 | b2));
end;

:: LATTICE8:def 8
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set
for b6 being set holds
      b6 = ConsecutiveDelta2(b4,b5)
   iff
      ex b7 being Relation-like Function-like T-Sequence-like set st
         b6 = last b7 &
          proj1 b7 = succ b5 &
          b7 . {} = b3 &
          (for b8 being ordinal set
                st succ b8 in succ b5
             holds b7 . succ b8 = new_bi_fun2(BiFun(b7 . b8,ConsecutiveSet2(b1,b8),b2),Quadr2(b4,b8))) &
          (for b8 being ordinal set
                st b8 in succ b5 & b8 <> {} & b8 is being_limit_ordinal
             holds b7 . b8 = union proj2 (b7 | b8));

:: LATTICE8:th 19
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3 holds
   ConsecutiveDelta2(b4,{}) = b3;

:: LATTICE8:th 20
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set holds
   ConsecutiveDelta2(b4,succ b5) = new_bi_fun2(BiFun(ConsecutiveDelta2(b4,b5),ConsecutiveSet2(b1,b5),b2),Quadr2(b4,b5));

:: LATTICE8:th 21
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being Relation-like Function-like T-Sequence-like set
for b6 being ordinal set
      st b6 <> {} &
         b6 is being_limit_ordinal &
         proj1 b5 = b6 &
         (for b7 being ordinal set
               st b7 in b6
            holds b5 . b7 = ConsecutiveDelta2(b4,b7))
   holds ConsecutiveDelta2(b4,b6) = union proj2 b5;

:: LATTICE8:th 22
theorem
for b1 being non empty set
for b2, b3, b4 being ordinal set
      st b3 c= b4
   holds ConsecutiveSet2(b1,b3) c= ConsecutiveSet2(b1,b4);

:: LATTICE8:th 23
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set holds
   ConsecutiveDelta2(b4,b5) is Function-like quasi_total Relation of [:ConsecutiveSet2(b1,b5),ConsecutiveSet2(b1,b5):],the carrier of b2;

:: LATTICE8:funcnot 6 => LATTICE8:func 6
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  let a4 be QuadrSeq of a3;
  let a5 be ordinal set;
  redefine func ConsecutiveDelta2(a4,a5) -> Function-like quasi_total Relation of [:ConsecutiveSet2(a1,a5),ConsecutiveSet2(a1,a5):],the carrier of a2;
end;

:: LATTICE8:th 24
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3
for b5 being ordinal set holds
   b3 c= ConsecutiveDelta2(b4,b5);

:: LATTICE8:th 25
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4, b5 being ordinal set
for b6 being QuadrSeq of b3
      st b4 c= b5
   holds ConsecutiveDelta2(b6,b4) c= ConsecutiveDelta2(b6,b5);

:: LATTICE8:th 26
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
   st b3 is zeroed(b1, b2)
for b4 being QuadrSeq of b3
for b5 being ordinal set holds
   ConsecutiveDelta2(b4,b5) is zeroed(ConsecutiveSet2(b1,b5), b2);

:: LATTICE8:th 27
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
   st b3 is symmetric(b1, b2)
for b4 being QuadrSeq of b3
for b5 being ordinal set holds
   ConsecutiveDelta2(b4,b5) is symmetric(ConsecutiveSet2(b1,b5), b2);

:: LATTICE8:th 28
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
   st b2 is modular
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
   st b3 is symmetric(b1, b2) & b3 is u.t.i.(b1, b2)
for b4 being ordinal set
for b5 being QuadrSeq of b3
      st b4 c= DistEsti b3
   holds ConsecutiveDelta2(b5,b4) is u.t.i.(ConsecutiveSet2(b1,b4), b2);

:: LATTICE8:th 29
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ordinal set
for b5 being QuadrSeq of b3
      st b4 c= DistEsti b3
   holds ConsecutiveDelta2(b5,b4) is Function-like quasi_total symmetric zeroed u.t.i. Relation of [:ConsecutiveSet2(b1,b4),ConsecutiveSet2(b1,b4):],the carrier of b2;

:: LATTICE8:funcnot 7 => LATTICE8:func 7
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  func NextSet2 A3 -> set equals
    ConsecutiveSet2(a1,DistEsti a3);
end;

:: LATTICE8:def 9
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2 holds
   NextSet2 b3 = ConsecutiveSet2(b1,DistEsti b3);

:: LATTICE8:funcreg 3
registration
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  cluster NextSet2 a3 -> non empty;
end;

:: LATTICE8:funcnot 8 => LATTICE8:func 8
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total Relation of [:a1,a1:],the carrier of a2;
  let a4 be QuadrSeq of a3;
  func NextDelta2 A4 -> set equals
    ConsecutiveDelta2(a4,DistEsti a3);
end;

:: LATTICE8:def 10
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total Relation of [:b1,b1:],the carrier of b2
for b4 being QuadrSeq of b3 holds
   NextDelta2 b4 = ConsecutiveDelta2(b4,DistEsti b3);

:: LATTICE8:funcnot 9 => LATTICE8:func 9
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr;
  let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
  let a4 be QuadrSeq of a3;
  redefine func NextDelta2 a4 -> Function-like quasi_total symmetric zeroed u.t.i. Relation of [:NextSet2 a3,NextSet2 a3:],the carrier of a2;
end;

:: LATTICE8:prednot 2 => LATTICE8:pred 2
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
  let a4 be non empty set;
  let a5 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a4,a4:],the carrier of a2;
  pred A4,A5 is_extension2_of A1,A3 means
    ex b1 being QuadrSeq of a3 st
       a4 = NextSet2 a3 & a5 = NextDelta2 b1;
end;

:: LATTICE8:dfs 10
definiens
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr;
  let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
  let a4 be non empty set;
  let a5 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a4,a4:],the carrier of a2;
To prove
     a4,a5 is_extension2_of a1,a3
it is sufficient to prove
  thus ex b1 being QuadrSeq of a3 st
       a4 = NextSet2 a3 & a5 = NextDelta2 b1;

:: LATTICE8:def 11
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being non empty set
for b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of b2 holds
      b4,b5 is_extension2_of b1,b3
   iff
      ex b6 being QuadrSeq of b3 st
         b4 = NextSet2 b3 & b5 = NextDelta2 b6;

:: LATTICE8:th 30
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being non empty set
for b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of b2
   st b4,b5 is_extension2_of b1,b3
for b6, b7 being Element of b1
for b8, b9 being Element of the carrier of b2
      st b3 .(b6,b7) <= b8 "\/" b9
   holds ex b10, b11 being Element of b4 st
      b5 .(b6,b10) = b8 &
       b5 .(b10,b11) = ((b3 .(b6,b7)) "\/" b8) "/\" b9 &
       b5 .(b11,b7) = b8;

:: LATTICE8:modenot 1 => LATTICE8:mode 1
definition
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr;
  let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
  mode ExtensionSeq2 of A1,A3 -> Relation-like Function-like set means
    proj1 it = NAT &
     it . 0 = [a1,a3] &
     (for b1 being Element of NAT holds
        ex b2 being non empty set st
           ex b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b2,b2:],the carrier of a2 st
              ex b4 being non empty set st
                 ex b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of a2 st
                    b4,b5 is_extension2_of b2,b3 & it . b1 = [b2,b3] & it . (b1 + 1) = [b4,b5]);
end;

:: LATTICE8:dfs 11
definiens
  let a1 be non empty set;
  let a2 be reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr;
  let a3 be Function-like quasi_total symmetric zeroed u.t.i. Relation of [:a1,a1:],the carrier of a2;
  let a4 be Relation-like Function-like set;
To prove
     a4 is ExtensionSeq2 of a1,a3
it is sufficient to prove
  thus proj1 a4 = NAT &
     a4 . 0 = [a1,a3] &
     (for b1 being Element of NAT holds
        ex b2 being non empty set st
           ex b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b2,b2:],the carrier of a2 st
              ex b4 being non empty set st
                 ex b5 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b4,b4:],the carrier of a2 st
                    b4,b5 is_extension2_of b2,b3 & a4 . b1 = [b2,b3] & a4 . (b1 + 1) = [b4,b5]);

:: LATTICE8:def 12
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being Relation-like Function-like set holds
      b4 is ExtensionSeq2 of b1,b3
   iff
      proj1 b4 = NAT &
       b4 . 0 = [b1,b3] &
       (for b5 being Element of NAT holds
          ex b6 being non empty set st
             ex b7 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b6,b6:],the carrier of b2 st
                ex b8 being non empty set st
                   ex b9 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b8,b8:],the carrier of b2 st
                      b8,b9 is_extension2_of b6,b7 & b4 . b5 = [b6,b7] & b4 . (b5 + 1) = [b8,b9]);

:: LATTICE8:th 31
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ExtensionSeq2 of b1,b3
for b5, b6 being Element of NAT
      st b5 <= b6
   holds (b4 . b5) `1 c= (b4 . b6) `1;

:: LATTICE8:th 32
theorem
for b1 being non empty set
for b2 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b3 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b1,b1:],the carrier of b2
for b4 being ExtensionSeq2 of b1,b3
for b5, b6 being Element of NAT
      st b5 <= b6
   holds (b4 . b5) `2 c= (b4 . b6) `2;

:: LATTICE8:th 33
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b2 being ExtensionSeq2 of the carrier of b1,BasicDF b1
for b3 being non empty set
      st b3 = union {(b2 . b4) `1 where b4 is Element of NAT: TRUE}
   holds union {(b2 . b4) `2 where b4 is Element of NAT: TRUE} is Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b3,b3:],the carrier of b1;

:: LATTICE8:th 34
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b2 being ExtensionSeq2 of the carrier of b1,BasicDF b1
for b3 being non empty set
for b4 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b3,b3:],the carrier of b1
for b5, b6 being Element of b3
for b7, b8 being Element of the carrier of b1
      st b3 = union {(b2 . b9) `1 where b9 is Element of NAT: TRUE} &
         b4 = union {(b2 . b9) `2 where b9 is Element of NAT: TRUE} &
         b4 .(b5,b6) <= b7 "\/" b8
   holds ex b9, b10 being Element of b3 st
      b4 .(b5,b9) = b7 &
       b4 .(b9,b10) = ((b4 .(b5,b6)) "\/" b7) "/\" b8 &
       b4 .(b10,b6) = b7;

:: LATTICE8:th 35
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr
for b2 being ExtensionSeq2 of the carrier of b1,BasicDF b1
for b3 being non empty set
for b4 being Function-like quasi_total symmetric zeroed u.t.i. Relation of [:b3,b3:],the carrier of b1
for b5 being Function-like quasi_total meet-preserving join-preserving Relation of the carrier of b1,the carrier of EqRelLATT b3
for b6, b7 being total symmetric transitive Relation of b3,b3
for b8, b9 being set
      st b5 = alpha b4 &
         b3 = union {(b2 . b10) `1 where b10 is Element of NAT: TRUE} &
         b4 = union {(b2 . b10) `2 where b10 is Element of NAT: TRUE} &
         b6 in the carrier of Image b5 &
         b7 in the carrier of Image b5 &
         [b8,b9] in b6 "\/" b7
   holds ex b10 being non empty FinSequence of b3 st
      len b10 = 2 + 2 & b8,b9 are_joint_by b10,b6,b7;

:: LATTICE8:th 36
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima modular RelStr holds
   b1 has_a_representation_of_type<= 2;

:: LATTICE8:th 37
theorem
for b1 being reflexive transitive antisymmetric lower-bounded with_suprema with_infima RelStr holds
      b1 has_a_representation_of_type<= 2
   iff
      b1 is modular;