Article GROEB_2, MML version 4.99.1005

:: GROEB_2:th 2
theorem
for b1 being non empty set
for b2 being FinSequence of b1
for b3, b4 being Element of NAT
      st b4 <= b3
   holds (b2 | b3) | b4 = b2 | b4;

:: GROEB_2:th 3
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3 being Element of NAT
      st for b4 being Element of NAT
              st b4 in dom b2 & b3 < b4
           holds b2 . b4 = 0. b1
   holds Sum b2 = Sum (b2 | b3);

:: GROEB_2:th 4
theorem
for b1 being non empty Abelian add-associative right_zeroed addLoopStr
for b2 being FinSequence of the carrier of b1
for b3, b4 being Element of NAT holds
Sum Swap(b2,b3,b4) = Sum b2;

:: GROEB_2:funcnot 1 => GROEB_2:func 1
definition
  let a1 be set;
  let a2, a3 be natural-valued finite-support ManySortedSet of a1;
  assume a3 divides a2;
  func A2 / A3 -> natural-valued finite-support ManySortedSet of a1 means
    a3 + it = a2;
end;

:: GROEB_2:def 1
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1
   st b3 divides b2
for b4 being natural-valued finite-support ManySortedSet of b1 holds
      b4 = b2 / b3
   iff
      b3 + b4 = b2;

:: GROEB_2:funcnot 2 => GROEB_2:func 2
definition
  let a1 be set;
  let a2, a3 be natural-valued finite-support ManySortedSet of a1;
  func lcm(A2,A3) -> natural-valued finite-support ManySortedSet of a1 means
    for b1 being set holds
       it . b1 = max(a2 . b1,a3 . b1);
  commutativity;
::  for a1 being set
::  for a2, a3 being natural-valued finite-support ManySortedSet of a1 holds
::  lcm(a2,a3) = lcm(a3,a2);
  idempotence;
::  for a1 being set
::  for a2 being natural-valued finite-support ManySortedSet of a1 holds
::     lcm(a2,a2) = a2;
end;

:: GROEB_2:def 2
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1 holds
   b4 = lcm(b2,b3)
iff
   for b5 being set holds
      b4 . b5 = max(b2 . b5,b3 . b5);

:: GROEB_2:funcnot 3 => GROEB_2:func 2
notation
  let a1 be set;
  let a2, a3 be natural-valued finite-support ManySortedSet of a1;
  synonym a2 lcm a3 for lcm(a2,a3);
end;

:: GROEB_2:prednot 1 => GROEB_2:pred 1
definition
  let a1 be set;
  let a2, a3 be natural-valued finite-support ManySortedSet of a1;
  pred A2,A3 are_disjoint means
    for b1 being set
          st a2 . b1 <> 0
       holds a3 . b1 = 0;
end;

:: GROEB_2:dfs 3
definiens
  let a1 be set;
  let a2, a3 be natural-valued finite-support ManySortedSet of a1;
To prove
     a2,a3 are_disjoint
it is sufficient to prove
  thus for b1 being set
          st a2 . b1 <> 0
       holds a3 . b1 = 0;

:: GROEB_2:def 3
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
   b2,b3 are_disjoint
iff
   for b4 being set
         st b2 . b4 <> 0
      holds b3 . b4 = 0;

:: GROEB_2:prednot 2 => not GROEB_2:pred 1
notation
  let a1 be set;
  let a2, a3 be natural-valued finite-support ManySortedSet of a1;
  antonym a2,a3 are_non_disjoint for a2,a3 are_disjoint;
end;

:: GROEB_2:th 7
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
b2 divides lcm(b2,b3) & b3 divides lcm(b2,b3);

:: GROEB_2:th 8
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
      st b2 divides b4 & b3 divides b4
   holds lcm(b2,b3) divides b4;

:: GROEB_2:th 9
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
   b2,b3 are_disjoint
iff
   lcm(b2,b3) = b2 + b3;

:: GROEB_2:th 10
theorem
for b1 being set
for b2 being natural-valued finite-support ManySortedSet of b1 holds
   b2 / b2 = EmptyBag b1;

:: GROEB_2:th 11
theorem
for b1 being set
for b2, b3 being natural-valued finite-support ManySortedSet of b1 holds
   b3 divides b2
iff
   lcm(b2,b3) = b2;

:: GROEB_2:th 12
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
      st b2 divides lcm(b3,b4)
   holds lcm(b3,b2) divides lcm(b3,b4);

:: GROEB_2:th 13
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
      st lcm(b3,b2) divides lcm(b3,b4)
   holds lcm(b2,b4) divides lcm(b3,b4);

:: GROEB_2:th 14
theorem
for b1 being set
for b2, b3, b4 being natural-valued finite-support ManySortedSet of b1
      st lcm(b2,b4) divides lcm(b3,b4)
   holds b2 divides lcm(b3,b4);

:: GROEB_2:th 15
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty Element of bool Bags b1 holds
   ex b4 being natural-valued finite-support ManySortedSet of b1 st
      b4 in b3 &
       (for b5 being natural-valued finite-support ManySortedSet of b1
             st b5 in b3
          holds b4 <= b5,b2);

:: GROEB_2:funcreg 1
registration
  let a1 be non trivial right_complementable add-associative right_zeroed addLoopStr;
  let a2 be non-zero Element of the carrier of a1;
  cluster - a2 -> non-zero;
end;

:: GROEB_2:funcreg 2
registration
  let a1 be set;
  let a2 be non empty add-cancelable distributive right_zeroed left_zeroed doubleLoopStr;
  let a3 be Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2;
  let a4 be Element of the carrier of a2;
  cluster a4 * a3 -> Function-like quasi_total monomial-like;
end;

:: GROEB_2:funcreg 3
registration
  let a1 be ordinal set;
  let a2 be non trivial add-cancelable distributive right_zeroed domRing-like left_zeroed doubleLoopStr;
  let a3 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a2;
  let a4 be non-zero Element of the carrier of a2;
  cluster a4 * a3 -> Function-like quasi_total non-zero;
end;

:: GROEB_2:th 16
theorem
for b1 being ordinal set
for b2 being non empty right-distributive right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b5 being natural-valued finite-support ManySortedSet of b1 holds
   b5 *' (b3 + b4) = (b5 *' b3) + (b5 *' b4);

:: GROEB_2:th 17
theorem
for b1 being ordinal set
for b2 being non empty right_complementable add-associative right_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being natural-valued finite-support ManySortedSet of b1 holds
   b4 *' - b3 = - (b4 *' b3);

:: GROEB_2:th 18
theorem
for b1 being ordinal set
for b2 being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being natural-valued finite-support ManySortedSet of b1
for b5 being Element of the carrier of b2 holds
   b4 *' (b5 * b3) = b5 * (b4 *' b3);

:: GROEB_2:th 19
theorem
for b1 being ordinal set
for b2 being non empty right-distributive doubleLoopStr
for b3, b4 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b5 being Element of the carrier of b2 holds
   b5 * (b3 + b4) = (b5 * b3) + (b5 * b4);

:: GROEB_2:th 20
theorem
for b1 being set
for b2 being non empty right_complementable add-associative right_zeroed doubleLoopStr
for b3 being Element of the carrier of b2 holds
   - (b3 |(b1,b2)) = (- b3) |(b1,b2);

:: GROEB_2:th 21
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
      st not 0_(b1,b3) in b4 &
         (for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
            st b5 <> b6 & b5 in b4 & b6 in b4
         for b7, b8 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
               st HM(b7 *' b5,b2) = HM(b8 *' b6,b2)
            holds PolyRedRel(b4,b2) reduces (b7 *' b5) - (b8 *' b6),0_(b1,b3))
   holds b4 is_Groebner_basis_wrt b2;

:: GROEB_2:funcnot 4 => GROEB_2:func 3
definition
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  func S-Poly(A4,A5,A2) -> Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 equals
    ((HC(a5,a2)) * (((lcm(HT(a4,a2),HT(a5,a2))) / HT(a4,a2)) *' a4)) - ((HC(a4,a2)) * (((lcm(HT(a4,a2),HT(a5,a2))) / HT(a5,a2)) *' a5));
end;

:: GROEB_2:def 4
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
S-Poly(b4,b5,b2) = ((HC(b5,b2)) * (((lcm(HT(b4,b2),HT(b5,b2))) / HT(b4,b2)) *' b4)) - ((HC(b4,b2)) * (((lcm(HT(b4,b2),HT(b5,b2))) / HT(b5,b2)) *' b5));

:: GROEB_2:th 22
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st b5 in b4 & b6 in b4
   holds S-Poly(b5,b6,b2) in b4 -Ideal;

:: GROEB_2:th 24
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3 holds
S-Poly(b4,b5,b2) = 0_(b1,b3);

:: GROEB_2:th 25
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   S-Poly(b4,0_(b1,b3),b2) = 0_(b1,b3) & S-Poly(0_(b1,b3),b4,b2) = 0_(b1,b3);

:: GROEB_2:th 26
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st S-Poly(b4,b5,b2) <> 0_(b1,b3)
   holds HT(S-Poly(b4,b5,b2),b2) < lcm(HT(b4,b2),HT(b5,b2)),b2;

:: GROEB_2:th 27
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
      st HT(b5,b2) divides HT(b4,b2)
   holds (HC(b5,b2)) * b4 top_reduces_to S-Poly(b4,b5,b2),b5,b2;

:: GROEB_2:th 28
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
   st b4 is_Groebner_basis_wrt b2
for b5, b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st b5 in b4 & b6 in b4 & b7 is_a_normal_form_of S-Poly(b5,b6,b2),PolyRedRel(b4,b2)
   holds b7 = 0_(b1,b3);

:: GROEB_2:th 29
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
   st for b5, b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
           st b5 in b4 & b6 in b4 & b7 is_a_normal_form_of S-Poly(b5,b6,b2),PolyRedRel(b4,b2)
        holds b7 = 0_(b1,b3)
for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st b5 in b4 & b6 in b4
   holds PolyRedRel(b4,b2) reduces S-Poly(b5,b6,b2),0_(b1,b3);

:: GROEB_2:th 30
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
      st not 0_(b1,b3) in b4 &
         (for b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
               st b5 in b4 & b6 in b4
            holds PolyRedRel(b4,b2) reduces S-Poly(b5,b6,b2),0_(b1,b3))
   holds b4 is_Groebner_basis_wrt b2;

:: GROEB_2:funcnot 5 => GROEB_2:func 4
definition
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Element of bool the carrier of Polynom-Ring(a1,a3);
  func S-Poly(A4,A2) -> Element of bool the carrier of Polynom-Ring(a1,a3) equals
    {S-Poly(b1,b2,a2) where b1 is Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3, b2 is Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3: b1 in a4 & b2 in a4};
end;

:: GROEB_2:def 5
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
   S-Poly(b4,b2) = {S-Poly(b5,b6,b2) where b5 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3, b6 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3: b5 in b4 & b6 in b4};

:: GROEB_2:funcreg 4
registration
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be finite Element of bool the carrier of Polynom-Ring(a1,a3);
  cluster S-Poly(a4,a2) -> finite;
end;

:: GROEB_2:th 31
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Element of bool the carrier of Polynom-Ring(b1,b3)
      st not 0_(b1,b3) in b4 &
         (for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
               st b5 in b4
            holds b5 is Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3)
   holds b4 is_Groebner_basis_wrt b2;

:: GROEB_2:th 32
theorem
for b1 being non empty multLoopStr
for b2 being non empty Element of bool the carrier of b1
for b3 being LeftLinearCombination of b2
for b4 being Element of NAT holds
   b3 | b4 is LeftLinearCombination of b2;

:: GROEB_2:th 33
theorem
for b1 being non empty multLoopStr
for b2 being non empty Element of bool the carrier of b1
for b3 being LeftLinearCombination of b2
for b4 being Element of NAT holds
   b3 /^ b4 is LeftLinearCombination of b2;

:: GROEB_2:th 34
theorem
for b1 being non empty multLoopStr
for b2, b3 being non empty Element of bool the carrier of b1
for b4 being LeftLinearCombination of b2
      st b2 c= b3
   holds b4 is LeftLinearCombination of b3;

:: GROEB_2:funcnot 6 => GROEB_2:func 5
definition
  let a1 be ordinal set;
  let a2 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a3 be non empty Element of bool the carrier of Polynom-Ring(a1,a2);
  let a4, a5 be LeftLinearCombination of a3;
  redefine func a4 ^ a5 -> LeftLinearCombination of a3;
end;

:: GROEB_2:prednot 3 => GROEB_2:pred 2
definition
  let a1 be ordinal set;
  let a2 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a3 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
  let a4 be non empty Element of bool the carrier of Polynom-Ring(a1,a2);
  let a5 be LeftLinearCombination of a4;
  pred A5 is_MonomialRepresentation_of A3 means
    Sum a5 = a3 &
     (for b1 being Element of NAT
           st b1 in dom a5
        holds ex b2 being Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2 st
           ex b3 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2 st
              b3 in a4 & a5 /. b1 = b2 *' b3);
end;

:: GROEB_2:dfs 6
definiens
  let a1 be ordinal set;
  let a2 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a3 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2;
  let a4 be non empty Element of bool the carrier of Polynom-Ring(a1,a2);
  let a5 be LeftLinearCombination of a4;
To prove
     a5 is_MonomialRepresentation_of a3
it is sufficient to prove
  thus Sum a5 = a3 &
     (for b1 being Element of NAT
           st b1 in dom a5
        holds ex b2 being Function-like quasi_total monomial-like Relation of Bags a1,the carrier of a2 st
           ex b3 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a2 st
              b3 in a4 & a5 /. b1 = b2 *' b3);

:: GROEB_2:def 6
theorem
for b1 being ordinal set
for b2 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b2)
for b5 being LeftLinearCombination of b4 holds
      b5 is_MonomialRepresentation_of b3
   iff
      Sum b5 = b3 &
       (for b6 being Element of NAT
             st b6 in dom b5
          holds ex b7 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b2 st
             ex b8 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2 st
                b8 in b4 & b5 /. b6 = b7 *' b8);

:: GROEB_2:th 35
theorem
for b1 being ordinal set
for b2 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b2)
for b5 being LeftLinearCombination of b4
      st b5 is_MonomialRepresentation_of b3
   holds Support b3 c= union {Support (b6 *' b7) where b6 is Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b2, b7 is Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2: ex b8 being Element of NAT st
      b8 in dom b5 & b5 /. b8 = b6 *' b7};

:: GROEB_2:th 36
theorem
for b1 being ordinal set
for b2 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b2)
for b6, b7 being LeftLinearCombination of b5
      st b6 is_MonomialRepresentation_of b3 & b7 is_MonomialRepresentation_of b4
   holds b6 ^ b7 is_MonomialRepresentation_of b3 + b4;

:: GROEB_2:prednot 4 => GROEB_2:pred 3
definition
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  let a6 be LeftLinearCombination of a5;
  let a7 be natural-valued finite-support ManySortedSet of a1;
  pred A6 is_Standard_Representation_of A4,A5,A7,A2 means
    Sum a6 = a4 &
     (for b1 being Element of NAT
           st b1 in dom a6
        holds ex b2 being Function-like quasi_total non-zero monomial-like Relation of Bags a1,the carrier of a3 st
           ex b3 being Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3 st
              b3 in a5 & a6 /. b1 = b2 *' b3 & HT(b2 *' b3,a2) <= a7,a2);
end;

:: GROEB_2:dfs 7
definiens
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  let a6 be LeftLinearCombination of a5;
  let a7 be natural-valued finite-support ManySortedSet of a1;
To prove
     a6 is_Standard_Representation_of a4,a5,a7,a2
it is sufficient to prove
  thus Sum a6 = a4 &
     (for b1 being Element of NAT
           st b1 in dom a6
        holds ex b2 being Function-like quasi_total non-zero monomial-like Relation of Bags a1,the carrier of a3 st
           ex b3 being Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a3 st
              b3 in a5 & a6 /. b1 = b2 *' b3 & HT(b2 *' b3,a2) <= a7,a2);

:: GROEB_2:def 7
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
for b7 being natural-valued finite-support ManySortedSet of b1 holds
      b6 is_Standard_Representation_of b4,b5,b7,b2
   iff
      Sum b6 = b4 &
       (for b8 being Element of NAT
             st b8 in dom b6
          holds ex b9 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3 st
             ex b10 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 st
                b10 in b5 & b6 /. b8 = b9 *' b10 & HT(b9 *' b10,b2) <= b7,b2);

:: GROEB_2:prednot 5 => GROEB_2:pred 4
definition
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  let a6 be LeftLinearCombination of a5;
  pred A6 is_Standard_Representation_of A4,A5,A2 means
    a6 is_Standard_Representation_of a4,a5,HT(a4,a2),a2;
end;

:: GROEB_2:dfs 8
definiens
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  let a6 be LeftLinearCombination of a5;
To prove
     a6 is_Standard_Representation_of a4,a5,a2
it is sufficient to prove
  thus a6 is_Standard_Representation_of a4,a5,HT(a4,a2),a2;

:: GROEB_2:def 8
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5 holds
      b6 is_Standard_Representation_of b4,b5,b2
   iff
      b6 is_Standard_Representation_of b4,b5,HT(b4,b2),b2;

:: GROEB_2:prednot 6 => GROEB_2:pred 5
definition
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  let a6 be natural-valued finite-support ManySortedSet of a1;
  pred A4 has_a_Standard_Representation_of A5,A6,A2 means
    ex b1 being LeftLinearCombination of a5 st
       b1 is_Standard_Representation_of a4,a5,a6,a2;
end;

:: GROEB_2:dfs 9
definiens
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  let a6 be natural-valued finite-support ManySortedSet of a1;
To prove
     a4 has_a_Standard_Representation_of a5,a6,a2
it is sufficient to prove
  thus ex b1 being LeftLinearCombination of a5 st
       b1 is_Standard_Representation_of a4,a5,a6,a2;

:: GROEB_2:def 9
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being natural-valued finite-support ManySortedSet of b1 holds
      b4 has_a_Standard_Representation_of b5,b6,b2
   iff
      ex b7 being LeftLinearCombination of b5 st
         b7 is_Standard_Representation_of b4,b5,b6,b2;

:: GROEB_2:prednot 7 => GROEB_2:pred 6
definition
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
  pred A4 has_a_Standard_Representation_of A5,A2 means
    ex b1 being LeftLinearCombination of a5 st
       b1 is_Standard_Representation_of a4,a5,a2;
end;

:: GROEB_2:dfs 10
definiens
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  let a3 be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be non empty Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
     a4 has_a_Standard_Representation_of a5,a2
it is sufficient to prove
  thus ex b1 being LeftLinearCombination of a5 st
       b1 is_Standard_Representation_of a4,a5,a2;

:: GROEB_2:def 10
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3) holds
      b4 has_a_Standard_Representation_of b5,b2
   iff
      ex b6 being LeftLinearCombination of b5 st
         b6 is_Standard_Representation_of b4,b5,b2;

:: GROEB_2:th 37
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
for b7 being natural-valued finite-support ManySortedSet of b1
      st b6 is_Standard_Representation_of b4,b5,b7,b2
   holds b6 is_MonomialRepresentation_of b4;

:: GROEB_2:th 38
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b7, b8 being LeftLinearCombination of b6
for b9 being natural-valued finite-support ManySortedSet of b1
      st b7 is_Standard_Representation_of b4,b6,b9,b2 & b8 is_Standard_Representation_of b5,b6,b9,b2
   holds b7 ^ b8 is_Standard_Representation_of b4 + b5,b6,b9,b2;

:: GROEB_2:th 39
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b7, b8 being LeftLinearCombination of b6
for b9 being natural-valued finite-support ManySortedSet of b1
for b10 being Element of NAT
      st b7 is_Standard_Representation_of b4,b6,b9,b2 & b8 = b7 | b10 & b5 = Sum (b7 /^ b10)
   holds b8 is_Standard_Representation_of b4 - b5,b6,b9,b2;

:: GROEB_2:th 40
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b7, b8 being LeftLinearCombination of b6
for b9 being natural-valued finite-support ManySortedSet of b1
for b10 being Element of NAT
      st b7 is_Standard_Representation_of b4,b6,b9,b2 & b8 = b7 /^ b10 & b5 = Sum (b7 | b10) & b10 <= len b7
   holds b8 is_Standard_Representation_of b4 - b5,b6,b9,b2;

:: GROEB_2:th 41
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
      st b6 is_MonomialRepresentation_of b4
   holds ex b7 being Element of NAT st
      ex b8 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3 st
         ex b9 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 st
            b7 in dom b6 & b9 in b5 & b6 . b7 = b8 *' b9 & HT(b4,b2) <= HT(b8 *' b9,b2),b2;

:: GROEB_2:th 42
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total Relation of Bags b1,Bags b1
for b3 being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b6 being LeftLinearCombination of b5
      st b6 is_Standard_Representation_of b4,b5,b2
   holds ex b7 being Element of NAT st
      ex b8 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3 st
         ex b9 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 st
            b9 in b5 & b7 in dom b6 & b6 /. b7 = b8 *' b9 & HT(b4,b2) = HT(b8 *' b9,b2);

:: GROEB_2:th 43
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
      st PolyRedRel(b5,b2) reduces b4,0_(b1,b3)
   holds b4 has_a_Standard_Representation_of b5,b2;

:: GROEB_2:th 44
theorem
for b1 being ordinal set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being non empty Element of bool the carrier of Polynom-Ring(b1,b3)
      st b4 has_a_Standard_Representation_of b5,b2
   holds b4 is_top_reducible_wrt b5,b2;

:: GROEB_2:th 45
theorem
for b1 being Element of NAT
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b3) holds
      b4 is_Groebner_basis_wrt b2
   iff
      for b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
            st b5 in b4 -Ideal
         holds b5 has_a_Standard_Representation_of b4,b2;