Article POLYRED, MML version 4.99.1005

:: POLYRED:exreg 1
registration
  let a1 be ordinal set;
  let a2 be non trivial ZeroStr;
  cluster Relation-like Function-like quasi_total finite-Support non-zero monomial-like Relation of Bags a1,the carrier of a2;
end;

:: POLYRED:exreg 2
registration
  cluster non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr;
end;

:: POLYRED:funcreg 1
registration
  let a1 be ordinal set;
  let a2 be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr;
  let a3, a4 be Function-like quasi_total finite-Support non-zero Relation of Bags a1,the carrier of a2;
  cluster a3 *' a4 -> Function-like quasi_total non-zero;
end;

:: POLYRED:th 1
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for b3, b4 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
- (b3 + b4) = (- b3) + - b4;

:: POLYRED:th 2
theorem
for b1 being set
for b2 being non empty left_zeroed addLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
   (0_(b1,b2)) + b3 = b3;

:: POLYRED:th 3
theorem
for b1 being 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 holds
   (- b3) + b3 = 0_(b1,b2) & b3 + - b3 = 0_(b1,b2);

:: POLYRED:th 4
theorem
for b1 being 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 holds
   b3 - 0_(b1,b2) = b3;

:: POLYRED:th 5
theorem
for b1 being ordinal set
for b2 being non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
   (0_(b1,b2)) *' b3 = 0_(b1,b2);

:: POLYRED:th 6
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b3, b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2 holds
- (b3 *' b4) = (- b3) *' b4 & - (b3 *' b4) = b3 *' - b4;

:: POLYRED:th 7
theorem
for b1 being ordinal set
for b2 being non empty right_complementable 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 Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b2
for b5 being natural-valued finite-support ManySortedSet of b1 holds
   (b4 *' b3) . ((term b4) + b5) = (b4 . term b4) * (b3 . b5);

:: POLYRED:th 8
theorem
for b1 being set
for b2 being non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2 holds
   (0. b2) * b3 = 0_(b1,b2);

:: POLYRED:th 9
theorem
for b1 being set
for b2 being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4 being Element of the carrier of b2 holds
   - (b4 * b3) = (- b4) * b3 & - (b4 * b3) = b4 * - b3;

:: POLYRED:th 10
theorem
for b1 being set
for b2 being non empty left-distributive doubleLoopStr
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4, b5 being Element of the carrier of b2 holds
(b4 * b3) + (b5 * b3) = (b4 + b5) * b3;

:: POLYRED:th 11
theorem
for b1 being set
for b2 being non empty associative multLoopStr_0
for b3 being Function-like quasi_total Relation of Bags b1,the carrier of b2
for b4, b5 being Element of the carrier of b2 holds
(b4 * b5) * b3 = b4 * (b5 * b3);

:: POLYRED:th 12
theorem
for b1 being ordinal set
for b2 being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed 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) = b3 *' (b5 * b4);

:: POLYRED:funcnot 1 => POLYRED:func 1
definition
  let a1 be ordinal set;
  let a2 be natural-valued finite-support ManySortedSet of a1;
  let a3 be non empty ZeroStr;
  let a4 be Function-like quasi_total Relation of Bags a1,the carrier of a3;
  func A2 *' A4 -> Function-like quasi_total Relation of Bags a1,the carrier of a3 means
    for b1 being natural-valued finite-support ManySortedSet of a1
          st a2 divides b1
       holds it . b1 = a4 . (b1 -' a2) &
        (for b2 being natural-valued finite-support ManySortedSet of a1
              st not a2 divides b2
           holds it . b2 = 0. a3);
end;

:: POLYRED:def 1
theorem
for b1 being ordinal set
for b2 being natural-valued finite-support ManySortedSet of b1
for b3 being non empty ZeroStr
for b4, b5 being Function-like quasi_total Relation of Bags b1,the carrier of b3 holds
   b5 = b2 *' b4
iff
   for b6 being natural-valued finite-support ManySortedSet of b1
         st b2 divides b6
      holds b5 . b6 = b4 . (b6 -' b2) &
       (for b7 being natural-valued finite-support ManySortedSet of b1
             st not b2 divides b7
          holds b5 . b7 = 0. b3);

:: POLYRED:funcreg 2
registration
  let a1 be ordinal set;
  let a2 be natural-valued finite-support ManySortedSet of a1;
  let a3 be non empty ZeroStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  cluster a2 *' a4 -> Function-like quasi_total finite-Support;
end;

:: POLYRED:th 13
theorem
for b1 being ordinal set
for b2, b3 being natural-valued finite-support ManySortedSet of b1
for b4 being non empty ZeroStr
for b5 being Function-like quasi_total Relation of Bags b1,the carrier of b4 holds
   (b2 *' b5) . (b3 + b2) = b5 . b3;

:: POLYRED:th 14
theorem
for b1 being ordinal set
for b2 being non empty ZeroStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being natural-valued finite-support ManySortedSet of b1 holds
   Support (b4 *' b3) c= {b4 + b5 where b5 is Element of Bags b1: b5 in Support b3};

:: POLYRED:th 15
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 ZeroStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being natural-valued finite-support ManySortedSet of b1 holds
   HT(b5 *' b4,b2) = b5 + HT(b4,b2);

:: POLYRED:th 16
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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5, b6 being natural-valued finite-support ManySortedSet of b1
      st b6 in Support (b5 *' b4)
   holds b6 <= b5 + HT(b4,b2),b2;

:: POLYRED:th 17
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 ZeroStr
for b4 being Function-like quasi_total Relation of Bags b1,the carrier of b3 holds
   (EmptyBag b1) *' b4 = b4;

:: POLYRED:th 18
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 ZeroStr
for b4 being Function-like quasi_total Relation of Bags b1,the carrier of b3
for b5, b6 being natural-valued finite-support ManySortedSet of b1 holds
(b5 + b6) *' b4 = b5 *' (b6 *' b4);

:: POLYRED:th 19
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable 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 Element of the carrier of b2 holds
   Support (b4 * b3) c= Support b3;

:: POLYRED:th 20
theorem
for b1 being ordinal set
for b2 being non trivial domRing-like doubleLoopStr
for b3 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b2
for b4 being non-zero Element of the carrier of b2 holds
   Support b3 c= Support (b4 * b3);

:: POLYRED:th 21
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 distributive add-associative right_zeroed domRing-like doubleLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b5 being non-zero Element of the carrier of b3 holds
   HT(b5 * b4,b2) = HT(b4,b2);

:: POLYRED:th 22
theorem
for b1 being ordinal set
for b2 being non trivial right_complementable distributive add-associative right_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
   b5 * (b4 *' b3) = (Monom(b5,b4)) *' b3;

:: POLYRED:th 23
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 well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3
for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
      st HT(b4,b2) in Support b5
   holds HT(b6 *' b4,b2) in Support (b6 *' b5);

:: POLYRED:funcreg 3
registration
  let a1 be ordinal set;
  let a2 be reflexive antisymmetric connected transitive total Relation of Bags a1,Bags a1;
  cluster RelStr(#Bags a1,a2#) -> strict connected;
end;

:: POLYRED:funcreg 4
registration
  let a1 be natural set;
  let a2 be reflexive antisymmetric transitive total admissible Relation of Bags a1,Bags a1;
  cluster RelStr(#Bags a1,a2#) -> strict well_founded;
end;

:: POLYRED:prednot 1 => POLYRED:pred 1
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 ZeroStr;
  let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  pred A4 <= A5,A2 means
    [Support a4,Support a5] in FinOrd RelStr(#Bags a1,a2#);
end;

:: POLYRED:dfs 2
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 ZeroStr;
  let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
     a4 <= a5,a2
it is sufficient to prove
  thus [Support a4,Support a5] in FinOrd RelStr(#Bags a1,a2#);

:: POLYRED:def 2
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 ZeroStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 <= b5,b2
iff
   [Support b4,Support b5] in FinOrd RelStr(#Bags b1,b2#);

:: POLYRED:prednot 2 => POLYRED:pred 2
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 ZeroStr;
  let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  pred A4 < A5,A2 means
    a4 <= a5,a2 & Support a4 <> Support a5;
end;

:: POLYRED:dfs 3
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 ZeroStr;
  let a4, a5 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
     a4 < a5,a2
it is sufficient to prove
  thus a4 <= a5,a2 & Support a4 <> Support a5;

:: POLYRED:def 3
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 ZeroStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 < b5,b2
iff
   b4 <= b5,b2 & Support b4 <> Support b5;

:: POLYRED:funcnot 2 => POLYRED:func 2
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 ZeroStr;
  let a4 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  func Support(A4,A2) -> Element of Fin the carrier of RelStr(#Bags a1,a2#) equals
    Support a4;
end;

:: POLYRED: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 empty ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   Support(b4,b2) = Support b4;

:: POLYRED: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 ZeroStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
   PosetMax Support(b4,b2) = HT(b4,b2);

:: POLYRED: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 empty addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 <= b4,b2;

:: POLYRED:th 26
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 addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 <= b5,b2 & b5 <= b4,b2
iff
   Support b4 = Support b5;

:: POLYRED: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 empty addLoopStr
for b4, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st b4 <= b5,b2 & b5 <= b6,b2
   holds b4 <= b6,b2;

:: POLYRED:th 28
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 addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st not b4 <= b5,b2
   holds b5 <= b4,b2;

:: POLYRED:th 29
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 addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 <= b5,b2
iff
   not b5 < b4,b2;

:: POLYRED:th 30
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 ZeroStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   0_(b1,b3) <= b4,b2;

:: POLYRED:th 31
theorem
for b1 being natural set
for b2 being reflexive antisymmetric connected transitive total admissible Relation of Bags b1,Bags b1
for b3 being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for b4 being non empty Element of bool the carrier of Polynom-Ring(b1,b3) holds
   ex b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
      b5 in b4 &
       (for b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
             st b6 in b4
          holds b5 <= b6,b2);

:: POLYRED:th 32
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 add-associative right_zeroed addLoopStr
for b4, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 < b5,b2
iff
   ((b4 = 0_(b1,b3) implies b5 = 0_(b1,b3)) & not HT(b4,b2) < HT(b5,b2),b2 implies HT(b4,b2) = HT(b5,b2) & Red(b4,b2) < Red(b5,b2),b2);

:: POLYRED:th 33
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 add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
   Red(b4,b2) < HM(b4,b2),b2;

:: POLYRED:th 34
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 add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   HM(b4,b2) <= b4,b2;

:: POLYRED:th 35
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 add-associative right_zeroed addLoopStr
for b4 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
   Red(b4,b2) < b4,b2;

:: POLYRED:prednot 3 => POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4, a5, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a7 be natural-valued finite-support ManySortedSet of a1;
  pred A4 reduces_to A6,A5,A7,A2 means
    a4 <> 0_(a1,a3) &
     a5 <> 0_(a1,a3) &
     a7 in Support a4 &
     (ex b1 being natural-valued finite-support ManySortedSet of a1 st
        b1 + HT(a5,a2) = a7 &
         a6 = a4 - (((a4 . a7) / HC(a5,a2)) * (b1 *' a5)));
end;

:: POLYRED:dfs 5
definiens
  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, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a7 be natural-valued finite-support ManySortedSet of a1;
To prove
     a4 reduces_to a6,a5,a7,a2
it is sufficient to prove
  thus a4 <> 0_(a1,a3) &
     a5 <> 0_(a1,a3) &
     a7 in Support a4 &
     (ex b1 being natural-valued finite-support ManySortedSet of a1 st
        b1 + HT(a5,a2) = a7 &
         a6 = a4 - (((a4 . a7) / HC(a5,a2)) * (b1 *' a5)));

:: POLYRED: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, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7 being natural-valued finite-support ManySortedSet of b1 holds
      b4 reduces_to b6,b5,b7,b2
   iff
      b4 <> 0_(b1,b3) &
       b5 <> 0_(b1,b3) &
       b7 in Support b4 &
       (ex b8 being natural-valued finite-support ManySortedSet of b1 st
          b8 + HT(b5,b2) = b7 &
           b6 = b4 - (((b4 . b7) / HC(b5,b2)) * (b8 *' b5)));

:: POLYRED:prednot 4 => POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4, a5, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  pred A4 reduces_to A6,A5,A2 means
    ex b1 being natural-valued finite-support ManySortedSet of a1 st
       a4 reduces_to a6,a5,b1,a2;
end;

:: POLYRED:dfs 6
definiens
  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, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
     a4 reduces_to a6,a5,a2
it is sufficient to prove
  thus ex b1 being natural-valued finite-support ManySortedSet of a1 st
       a4 reduces_to a6,a5,b1,a2;

:: POLYRED:def 6
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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 reduces_to b6,b5,b2
iff
   ex b7 being natural-valued finite-support ManySortedSet of b1 st
      b4 reduces_to b6,b5,b7,b2;

:: POLYRED:prednot 5 => POLYRED: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 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;
  let a6 be Element of bool the carrier of Polynom-Ring(a1,a3);
  pred A4 reduces_to A5,A6,A2 means
    ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       b1 in a6 & a4 reduces_to a5,b1,a2;
end;

:: POLYRED: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 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;
  let a6 be Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
     a4 reduces_to a5,a6,a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       b1 in a6 & a4 reduces_to a5,b1,a2;

:: POLYRED: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 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
for b6 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
      b4 reduces_to b5,b6,b2
   iff
      ex b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
         b7 in b6 & b4 reduces_to b5,b7,b2;

:: POLYRED:prednot 6 => POLYRED: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 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;
  pred A4 is_reducible_wrt A5,A2 means
    ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       a4 reduces_to b1,a5,a2;
end;

:: POLYRED: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 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;
To prove
     a4 is_reducible_wrt a5,a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       a4 reduces_to b1,a5,a2;

:: POLYRED: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 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
   b4 is_reducible_wrt b5,b2
iff
   ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
      b4 reduces_to b6,b5,b2;

:: POLYRED:prednot 7 => not POLYRED:pred 6
notation
  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;
  antonym a4 is_irreducible_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;

:: POLYRED:prednot 8 => not POLYRED:pred 6
notation
  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;
  antonym a4 is_in_normalform_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;

:: POLYRED:prednot 9 => POLYRED:pred 7
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
  pred A4 is_reducible_wrt A5,A2 means
    ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       a4 reduces_to b1,a5,a2;
end;

:: POLYRED: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 trivial right_complementable almost_left_invertible associative commutative 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 Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
     a4 is_reducible_wrt a5,a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       a4 reduces_to b1,a5,a2;

:: POLYRED: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 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
for b5 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
      b4 is_reducible_wrt b5,b2
   iff
      ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
         b4 reduces_to b6,b5,b2;

:: POLYRED:prednot 10 => not POLYRED:pred 7
notation
  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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
  antonym a4 is_irreducible_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;

:: POLYRED:prednot 11 => not POLYRED:pred 7
notation
  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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
  antonym a4 is_in_normalform_wrt a5,a2 for a4 is_reducible_wrt a5,a2;
end;

:: POLYRED:prednot 12 => POLYRED:pred 8
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, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  pred A4 top_reduces_to A6,A5,A2 means
    a4 reduces_to a6,a5,HT(a4,a2),a2;
end;

:: POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr;
  let a4, a5, a6 be Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
To prove
     a4 top_reduces_to a6,a5,a2
it is sufficient to prove
  thus a4 reduces_to a6,a5,HT(a4,a2),a2;

:: POLYRED: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 trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
   b4 top_reduces_to b6,b5,b2
iff
   b4 reduces_to b6,b5,HT(b4,b2),b2;

:: POLYRED:prednot 13 => POLYRED:pred 9
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;
  pred A4 is_top_reducible_wrt A5,A2 means
    ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       a4 top_reduces_to b1,a5,a2;
end;

:: POLYRED:dfs 11
definiens
  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;
To prove
     a4 is_top_reducible_wrt a5,a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       a4 top_reduces_to b1,a5,a2;

:: POLYRED:def 11
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
   b4 is_top_reducible_wrt b5,b2
iff
   ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
      b4 top_reduces_to b6,b5,b2;

:: POLYRED:prednot 14 => POLYRED:pred 10
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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
  pred A4 is_top_reducible_wrt A5,A2 means
    ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       b1 in a5 & a4 is_top_reducible_wrt b1,a2;
end;

:: POLYRED:dfs 12
definiens
  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 Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3;
  let a5 be Element of bool the carrier of Polynom-Ring(a1,a3);
To prove
     a4 is_top_reducible_wrt a5,a2
it is sufficient to prove
  thus ex b1 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 st
       b1 in a5 & a4 is_top_reducible_wrt b1,a2;

:: POLYRED:def 12
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
for b5 being Element of bool the carrier of Polynom-Ring(b1,b3) holds
      b4 is_top_reducible_wrt b5,b2
   iff
      ex b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
         b6 in b5 & b4 is_top_reducible_wrt b6,b2;

:: POLYRED:th 36
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
for b5 being Function-like quasi_total finite-Support non-zero Relation of Bags b1,the carrier of b3 holds
      b4 is_reducible_wrt b5,b2
   iff
      ex b6 being natural-valued finite-support ManySortedSet of b1 st
         b6 in Support b4 & HT(b5,b2) divides b6;

:: POLYRED: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 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
   0_(b1,b3) is_irreducible_wrt b4,b2;

:: POLYRED:th 38
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, b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
      st b4 reduces_to b4 - (b6 *' b5),b5,b2
   holds HT(b6 *' b5,b2) in Support b4;

:: POLYRED: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 degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for b4, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7 being natural-valued finite-support ManySortedSet of b1
      st b4 reduces_to b6,b5,b7,b2
   holds not b7 in Support b6;

:: POLYRED:th 40
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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7, b8 being natural-valued finite-support ManySortedSet of b1
      st b7 < b8,b2 & b4 reduces_to b6,b5,b7,b2
   holds    b8 in Support b6
   iff
      b8 in Support b4;

:: POLYRED:th 41
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, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b7, b8 being natural-valued finite-support ManySortedSet of b1
      st b7 < b8,b2 & b4 reduces_to b6,b5,b7,b2
   holds b4 . b8 = b6 . b8;

:: POLYRED:th 42
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 add-associative right_zeroed doubleLoopStr
for b4, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
   st b4 reduces_to b6,b5,b2
for b7 being natural-valued finite-support ManySortedSet of b1
      st b7 in Support b6
   holds b7 <= HT(b4,b2),b2;

:: POLYRED: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, b5, b6 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st b4 reduces_to b6,b5,b2
   holds b6 < b4,b2;

:: POLYRED:funcnot 3 => POLYRED: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 be Element of bool the carrier of Polynom-Ring(a1,a3);
  func PolyRedRel(A4,A2) -> Relation of (the carrier of Polynom-Ring(a1,a3)) \ {0_(a1,a3)},the carrier of Polynom-Ring(a1,a3) means
    for b1, b2 being Function-like quasi_total finite-Support Relation of Bags a1,the carrier of a3 holds
       [b1,b2] in it
    iff
       b1 reduces_to b2,a4,a2;
end;

:: POLYRED:def 13
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)
for b5 being Relation of (the carrier of Polynom-Ring(b1,b3)) \ {0_(b1,b3)},the carrier of Polynom-Ring(b1,b3) holds
      b5 = PolyRedRel(b4,b2)
   iff
      for b6, b7 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 holds
         [b6,b7] in b5
      iff
         b6 reduces_to b7,b4,b2;

:: POLYRED: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 empty non degenerated right_complementable almost_left_invertible associative commutative 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 Element of bool the carrier of Polynom-Ring(b1,b3)
      st PolyRedRel(b6,b2) reduces b4,b5
   holds b5 <= b4,b2 &
    (b5 = 0_(b1,b3) or HT(b5,b2) <= HT(b4,b2),b2);

:: POLYRED:funcreg 5
registration
  let a1 be natural set;
  let a2 be reflexive antisymmetric connected transitive total admissible Relation of Bags a1,Bags a1;
  let a3 be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr;
  let a4 be Element of bool the carrier of Polynom-Ring(a1,a3);
  cluster PolyRedRel(a4,a2) -> strongly-normalizing;
end;

:: POLYRED:th 45
theorem
for b1 being natural 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 Abelian add-associative right_zeroed left_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
   holds PolyRedRel(b4,b2) reduces b6 *' b5,0_(b1,b3);

:: POLYRED:th 46
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 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
for b7 being Function-like quasi_total non-zero monomial-like Relation of Bags b1,the carrier of b3
      st b5 reduces_to b6,b4,b2
   holds b7 *' b5 reduces_to b7 *' b6,b4,b2;

:: POLYRED:th 47
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 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
for b7 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
      st PolyRedRel(b4,b2) reduces b5,b6
   holds PolyRedRel(b4,b2) reduces b7 *' b5,b7 *' b6;

:: POLYRED:th 48
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 Element of bool the carrier of Polynom-Ring(b1,b3)
for b5 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
for b6 being Function-like quasi_total monomial-like Relation of Bags b1,the carrier of b3
      st PolyRedRel(b4,b2) reduces b5,0_(b1,b3)
   holds PolyRedRel(b4,b2) reduces b6 *' b5,0_(b1,b3);

:: POLYRED:th 49
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, b7, b8 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st b5 - b6 = b7 & PolyRedRel(b4,b2) reduces b7,b8
   holds ex b9, b10 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3 st
      b9 - b10 = b8 & PolyRedRel(b4,b2) reduces b5,b9 & PolyRedRel(b4,b2) reduces b6,b10;

:: POLYRED:th 50
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 PolyRedRel(b4,b2) reduces b5 - b6,0_(b1,b3)
   holds b5,b6 are_convergent_wrt PolyRedRel(b4,b2);

:: POLYRED:th 51
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 PolyRedRel(b4,b2) reduces b5 - b6,0_(b1,b3)
   holds b5,b6 are_convertible_wrt PolyRedRel(b4,b2);

:: POLYRED:prednot 15 => POLYRED:pred 11
definition
  let a1 be non empty addLoopStr;
  let a2 be Element of bool the carrier of a1;
  let a3, a4 be Element of the carrier of a1;
  pred A3,A4 are_congruent_mod A2 means
    a3 - a4 in a2;
end;

:: POLYRED:dfs 14
definiens
  let a1 be non empty addLoopStr;
  let a2 be Element of bool the carrier of a1;
  let a3, a4 be Element of the carrier of a1;
To prove
     a3,a4 are_congruent_mod a2
it is sufficient to prove
  thus a3 - a4 in a2;

:: POLYRED:def 14
theorem
for b1 being non empty addLoopStr
for b2 being Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1 holds
   b3,b4 are_congruent_mod b2
iff
   b3 - b4 in b2;

:: POLYRED:th 52
theorem
for b1 being non empty right_complementable right-distributive add-associative right_zeroed left_zeroed doubleLoopStr
for b2 being non empty right-ideal Element of bool the carrier of b1
for b3 being Element of the carrier of b1 holds
   b3,b3 are_congruent_mod b2;

:: POLYRED:th 53
theorem
for b1 being non empty right_complementable right-distributive well-unital add-associative right_zeroed doubleLoopStr
for b2 being non empty right-ideal Element of bool the carrier of b1
for b3, b4 being Element of the carrier of b1
      st b3,b4 are_congruent_mod b2
   holds b4,b3 are_congruent_mod b2;

:: POLYRED:th 54
theorem
for b1 being non empty right_complementable add-associative right_zeroed addLoopStr
for b2 being non empty add-closed Element of bool the carrier of b1
for b3, b4, b5 being Element of the carrier of b1
      st b3,b4 are_congruent_mod b2 & b4,b5 are_congruent_mod b2
   holds b3,b5 are_congruent_mod b2;

:: POLYRED:th 55
theorem
for b1 being non trivial right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for b2 being non empty add-closed Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
      st b3,b4 are_congruent_mod b2 & b5,b6 are_congruent_mod b2
   holds b3 + b5,b4 + b6 are_congruent_mod b2;

:: POLYRED:th 56
theorem
for b1 being non empty right_complementable commutative distributive add-associative right_zeroed doubleLoopStr
for b2 being non empty add-closed right-ideal Element of bool the carrier of b1
for b3, b4, b5, b6 being Element of the carrier of b1
      st b3,b4 are_congruent_mod b2 & b5,b6 are_congruent_mod b2
   holds b3 * b5,b4 * b6 are_congruent_mod b2;

:: POLYRED:th 57
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 Element of the carrier of Polynom-Ring(b1,b3)
      st b5,b6 are_convertible_wrt PolyRedRel(b4,b2)
   holds b5,b6 are_congruent_mod b4 -Ideal;

:: POLYRED:th 58
theorem
for b1 being natural 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 non empty Element of bool the carrier of Polynom-Ring(b1,b3)
for b5, b6 being Element of the carrier of Polynom-Ring(b1,b3)
      st b5,b6 are_congruent_mod b4 -Ideal
   holds b5,b6 are_convertible_wrt PolyRedRel(b4,b2);

:: POLYRED:th 59
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 PolyRedRel(b4,b2) reduces b5,b6
   holds b5 - b6 in b4 -Ideal;

:: POLYRED:th 60
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 being Function-like quasi_total finite-Support Relation of Bags b1,the carrier of b3
      st PolyRedRel(b4,b2) reduces b5,0_(b1,b3)
   holds b5 in b4 -Ideal;