
theorem Th30:
  for n being Element of NAT, T being connected admissible
  TermOrder of n, L being add-associative right_complementable right_zeroed
commutative associative well-unital distributive Abelian almost_left_invertible
  non trivial doubleLoopStr holds {0_(n,L)} is_Groebner_basis_of {0_(n,L)},T
proof
  let n be Element of NAT, T be connected admissible TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non trivial
  doubleLoopStr;
  set I = {0_(n,L)}, G = {0_(n,L)}, R = PolyRedRel(G,T);
A1: 0_(n,L) = 0.Polynom-Ring(n,L) by POLYNOM1:def 11;
  now
    let a,b,c be object;
    assume that
A2: [a,b] in R and
    [a,c] in R;
    consider p,q being object such that
A3: p in NonZero Polynom-Ring(n,L) and
A4: q in the carrier of Polynom-Ring(n,L) and
A5: [a,b] = [p,q] by A2,ZFMISC_1:def 2;
    reconsider q as Polynomial of n,L by A4,POLYNOM1:def 11;
    not p in {0_(n,L)} by A1,A3,XBOOLE_0:def 5;
    then p <> 0_(n,L) by TARSKI:def 1;
    then reconsider p as non-zero Polynomial of n,L by A3,POLYNOM1:def 11
,POLYNOM7:def 1;
    p reduces_to q,G,T by A2,A5,POLYRED:def 13;
    then consider g being Polynomial of n,L such that
A6: g in G and
A7: p reduces_to q,g,T by POLYRED:def 7;
    g = 0_(n,L) by A6,TARSKI:def 1;
    then p is_reducible_wrt 0_(n,L),T by A7,POLYRED:def 8;
    hence b,c are_convergent_wrt R by Lm3;
  end;
  then
A8: PolyRedRel(G,T) is locally-confluent by REWRITE1:def 24;
  0_(n,L) = 0.(Polynom-Ring(n,L)) by POLYNOM1:def 11;
  then G-Ideal = I by IDEAL_1:44;
  hence thesis by A8;
end;
