
theorem
  for n being Element of NAT, T being connected admissible TermOrder of
  n, L being Abelian add-associative right_complementable right_zeroed
  commutative associative well-unital distributive almost_left_invertible non
degenerated non empty doubleLoopStr, I being add-closed left-ideal non empty
  Subset of Polynom-Ring(n,L) st I <> {0_(n,L)} ex G being finite Subset of
  Polynom-Ring(n,L) st G is_Groebner_basis_of I,T & not(0_(n,L) in G)
proof
  let n be Element of NAT, T be connected admissible TermOrder of n, L be
  Abelian add-associative right_complementable right_zeroed commutative
  associative well-unital distributive almost_left_invertible non degenerated
  non empty doubleLoopStr, I be add-closed left-ideal non empty Subset of
  Polynom-Ring(n,L);
  assume
A1: I <> {0_(n,L)};
A2: 0_(n,L) = 0.Polynom-Ring(n,L) by POLYNOM1:def 11;
  consider G being finite Subset of Polynom-Ring(n,L) such that
A3: G is_Groebner_basis_of I,T by Th35;
  set R = PolyRedRel(G,T);
A4: G-Ideal = I by A3;
A5: PolyRedRel(G,T) is locally-confluent by A3;
    set G9 = G \ {0_(n,L)}, R9 = PolyRedRel(G9,T);
    reconsider G9 as finite Subset of Polynom-Ring(n,L);
A6: now
      per cases;
      case
A7:     G9 = {};
        now
          per cases;
          case
            G = {};
            hence G9-Ideal = I by A3;
          end;
          case
A8:         G <> {};
A9:        now
              let u be object;
              assume u in {0_(n,L)};
              then
A10:          u = 0_(n,L) by TARSKI:def 1;
A11:          G c= {0_(n,L)} by A7,XBOOLE_1:37;
              now
                assume not u in G;
            then for v being object holds not v in G by A10,A11,TARSKI:def 1;
                hence G = {} by XBOOLE_0:def 1;
              end;
              hence u in G by A8;
            end;
A12:        0_(n,L) = 0.(Polynom-Ring(n,L)) by POLYNOM1:def 11;
            now
              let u be object;
              assume
A13:          u in G;
              G c= {0_(n,L)} by A7,XBOOLE_1:37;
              hence u in {0_(n,L)} by A13;
            end;
            then G = {0_(n,L)} by A9,TARSKI:2;
            hence G9-Ideal = I by A1,A4,A12,IDEAL_1:44;
          end;
        end;
        hence G9-Ideal = I;
      end;
      case
        G9 <> {};
        then reconsider
        GG = G,GG9 = G9 as non empty Subset of Polynom-Ring(n,L);
A14:    0.(Polynom-Ring(n,L)) = 0_(n,L) by POLYNOM1:def 11;
A15:    now
          let u be object;
          assume u in G-Ideal;
          then ex f being LinearCombination of GG st u = Sum f by IDEAL_1:60;
          then ex g being LinearCombination of GG9 st u = Sum g by A14,Lm9;
          hence u in G9-Ideal by IDEAL_1:60;
        end;
        now
          let u be object;
A16:      GG9-Ideal c= GG-Ideal by IDEAL_1:57,XBOOLE_1:36;
          assume u in G9-Ideal;
          hence u in G-Ideal by A16;
        end;
        hence G9-Ideal = I by A4,A15,TARSKI:2;
      end;
    end;
A17: now
      assume 0_(n,L) in G9;
      then not(0_(n,L)) in {0_(n,L)} by XBOOLE_0:def 5;
      hence contradiction by TARSKI:def 1;
    end;
A18: for u being object holds u in R implies u in R9
    proof
      let u be object;
      assume
A19:  u in R;
      then consider p,q being object such that
A20:  p in NonZero Polynom-Ring(n,L) and
A21:  q in the carrier of Polynom-Ring(n,L) and
A22:  u = [p,q] by ZFMISC_1:def 2;
      reconsider q as Polynomial of n,L by A21,POLYNOM1:def 11;
      not p in {0_(n,L)} by A2,A20,XBOOLE_0:def 5;
      then p <> 0_(n,L) by TARSKI:def 1;
      then reconsider p as non-zero Polynomial of n,L by A20,POLYNOM1:def 11
,POLYNOM7:def 1;
      p reduces_to q,G,T by A19,A22,POLYRED:def 13;
      then consider f being Polynomial of n,L such that
A23:  f in G and
A24:  p reduces_to q,f,T by POLYRED:def 7;
      ex b being bag of n st p reduces_to q,f,b,T by A24,POLYRED:def 6;
      then f <> 0_(n,L) by POLYRED:def 5;
      then not f in {0_(n,L)} by TARSKI:def 1;
      then f in G9 by A23,XBOOLE_0:def 5;
      then p reduces_to q,G9,T by A24,POLYRED:def 7;
      hence thesis by A22,POLYRED:def 13;
    end;
    R9 c= R by Th4,XBOOLE_1:36;
    then for u being object holds u in R9 implies u in R;
    then R9 is locally-confluent by A5,A18,TARSKI:2;
    then G9 is_Groebner_basis_of I,T by A6;
    hence thesis by A17;
end;
