
theorem
  for R being Abelian right_zeroed add-associative right_complementable
  well-unital distributive associative commutative non trivial
  doubleLoopStr, X be infinite Ordinal holds Polynom-Ring (X,R) is non
  Noetherian
proof
  deffunc F(set)=$1;
  let R be Abelian right_zeroed add-associative right_complementable
  well-unital distributive associative commutative non trivial
  doubleLoopStr, X be infinite Ordinal such that
A1: Polynom-Ring (X,R) is Noetherian;
  reconsider f0 = X --> 0.R as Function of X,the carrier of R;
  deffunc G(Element of X)=1_1($1,R);
  set tcR = the carrier of R;
A2: for d1,d2 being Element of X st G(d1)=G(d2) holds d1=d2 by Th14;
  tcR c= tcR;
  then reconsider cR = the carrier of R as non empty Subset of tcR;
  set S = {1_1(n, R) where n is Element of X : n in NAT};
  set tcPR = the carrier of Polynom-Ring(X,R);
A3: NAT c= X by CARD_3:85;
  reconsider 0X = 0 as Element of X by A3;
A4: S c= tcPR
  proof
    let x be object;
    assume x in S;
    then ex n being Element of X st x = 1_1(n,R) & n in NAT;
    hence thesis by POLYNOM1:def 11;
  end;
  1_1(0X,R) in S;
  then reconsider S as non empty Subset of tcPR by A4;
  consider C being non empty finite Subset of tcPR such that
A5: C c= S and
A6: C-Ideal = S-Ideal by A1,IDEAL_1:96;
  set CN = { F(n) where n is Element of X : G(n) in C };
A7: C is finite;
A8: CN is finite from FUNCT_7:sch 2(A7,A2);
A9: CN c= NAT
  proof
    let x be object;
    assume x in CN;
    then consider n being Element of X such that
A10: x = n and
A11: 1_1(n,R) in C;
    1_1(n,R) in S by A5,A11;
    then ex m being Element of X st 1_1(n,R) = 1_1(m,R) & m in NAT;
    hence thesis by A10,Th14;
  end;
  set c = the Element of C;
  c in S by A5;
  then consider cn being Element of X such that
A12: c = 1_1(cn,R) and
A13: cn in NAT;
  reconsider cn as Element of NAT by A13;
  cn in CN by A12;
  then reconsider CN as non empty finite Subset of NAT by A8,A9;
  reconsider mm = max CN as Element of NAT by ORDINAL1:def 12;
  reconsider m1 = mm+1 as Element of NAT;
  reconsider m2 = m1 as Element of X by A3;
  1_1(m2,R) in S & S c= S-Ideal by IDEAL_1:def 14;
  then consider lc being LinearCombination of C such that
A14: 1_1(m2,R) = Sum lc by A6,IDEAL_1:60;
  reconsider ev = f0+*(m2,1_R) as Function of X, R;
  consider E be FinSequence of [:tcPR, tcPR, tcPR:] such that
A15: E represents lc by IDEAL_1:35;
  set P = Polynom-Evaluation(X, R, ev);
  deffunc F(Nat)=(P.((E/.$1)`1_3))*(P.((E/.$1)`2_3))*(P.((E/.$1)`3_3));
  consider LC being FinSequence of the carrier of R such that
A16: len LC = len lc and
A17: for k being Nat st k in dom LC holds LC.k = F(k) from FINSEQ_2:sch
  1;
  now
    let i being set;
    assume
A18: i in dom LC;
    then reconsider k = i as Element of NAT;
    reconsider a = (P.((E/.k)`2_3)) as Element of cR;
    reconsider u=(P.((E/.k)`1_3)), v=(P.((E/.k)`3_3)) as Element of R;
    take u, v, a;
    thus LC/.i = LC.k by A18,PARTFUN1:def 6
      .= u*a*v by A17,A18;
  end;
  then reconsider LC as LinearCombination of cR by IDEAL_1:def 8;
A19: now
    let i be Element of NAT;
A20: now
      assume m2 in CN;
      then (max CN)+1 <= max CN by XXREAL_2:def 8;
      hence contradiction by XREAL_1:29;
    end;
    assume
A21: i in dom LC;
    then i in dom lc by A16,FINSEQ_3:29;
    then reconsider y = (E/.i)`2_3 as Element of C by A15;
    y in S by A5;
    then consider n being Element of X such that
A22: y = 1_1(n, R) and
    n in NAT;
    n in CN by A22;
    then
A23: ev.n = (X --> 0.R).n by A20,FUNCT_7:32
      .= 0.R;
A24: Support(1_1(n,R)) = {UnitBag n} by Th13;
A25: P.1_1(n,R) = eval(1_1(n,R), ev) by POLYNOM2:def 5
      .= (1_1(n,R).UnitBag n)*eval(UnitBag n,ev) by A24,POLYNOM2:19
      .= 1_R *eval(UnitBag n,ev) by Th12
      .= 1_R * ev.n by Th11
      .= 0.R by A23;
    thus LC.i = (P.((E/.i)`1_3))*(P.((E/.i)`2_3))*(P.((E/.i)`3_3))
               by A17,A21
      .= 0.R*(P.((E/.i)`3_3)) by A22,A25
      .= 0.R;
  end;
  dom (X --> 0.R) = X;
  then
A26: ev.m2 = 1_R by FUNCT_7:31;
A27: Support(1_1(m2,R)) = {UnitBag m2} by Th13;
A28: Polynom-Evaluation(X, R, ev).1_1(m2,R) = eval(1_1(m2,R), ev) by
POLYNOM2:def 5
    .= (1_1(m2,R).UnitBag m2)*eval(UnitBag m2,ev) by A27,POLYNOM2:19
    .= 1_R *eval(UnitBag m2,ev) by Th12
    .= 1_R * ev.m2 by Th11
    .= 1_R by A26;
  for k being set st k in dom LC
    holds LC.k = (P.((E/.k)`1_3))*(P.((E/.k)`2_3))*(P.((E/.k)`3_3)) by A17;
  then P.(Sum lc) = Sum LC by A15,A16,Th24
    .=0.R by A19,POLYNOM3:1;
  hence contradiction by A14,A28;
end;
