
theorem
  for T being admissible connected TermOrder of {}, L being
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non degenerated non
  empty doubleLoopStr, I being add-closed left-ideal non empty Subset of
Polynom-Ring({},L), P being non empty Subset of Polynom-Ring({},L) st P c= I &
  P <> {0_({},L)} holds P is_Groebner_basis_of I,T
proof
  now
    set j = the Element of {i where i is Element of NAT: i < 0};
    assume {i where i is Element of NAT: i < 0} <> {};
    then j in {i where i is Element of NAT: i < 0};
    then ex i being Element of NAT st i = j & i < 0;
    hence contradiction;
  end;
  then reconsider n = {} as Element of NAT;
  let T be admissible connected TermOrder of {}, L be add-associative
  right_complementable right_zeroed commutative associative well-unital
  distributive Abelian almost_left_invertible non degenerated non empty
doubleLoopStr, I be add-closed left-ideal non empty Subset of Polynom-Ring({},
  L), P be non empty Subset of Polynom-Ring({},L);
  assume that
A1: P c= I and
A2: P <> {0_({},L)};
  reconsider T as admissible connected TermOrder of n;
  reconsider P as non empty Subset of Polynom-Ring(n,L);
  reconsider I as add-closed left-ideal non empty Subset of Polynom-Ring(n,L);
A3: ex q being Element of P st q <> 0_(n,L)
  proof
    assume
A4: not(ex q being Element of P st q <> 0_(n,L));
A5: now
      let u be object;
      assume u in {0_(n,L)};
      then
A6:   u = 0_(n,L) by TARSKI:def 1;
      now
        assume not u in P;
        then for v being object holds not v in P by A4,A6;
        hence thesis by XBOOLE_0:def 1;
      end;
      hence u in P;
    end;
    now
      let u be object;
      assume u in P;
      then u = 0_(n,L) by A4;
      hence u in {0_(n,L)} by TARSKI:def 1;
    end;
    hence thesis by A2,A5,TARSKI:2;
  end;
  now
    consider p being Element of P such that
A7: p <> 0_(n,L) by A3;
    reconsider p as Polynomial of n,L by POLYNOM1:def 11;
    reconsider p as non-zero Polynomial of n,L by A7,POLYNOM7:def 1;
    let f be non-zero Polynomial of n,L;
    assume f in I;
    f <> 0_(n,L) by POLYNOM7:def 1;
    then Support f <> {} by POLYNOM7:1;
    then HT(f,T) in Support f by TERMORD:def 6;
    then HT(p,T) = EmptyBag n & EmptyBag n in Support f;
    then f is_reducible_wrt p,T by POLYRED:36;
    then consider g being Polynomial of n,L such that
A8: f reduces_to g,p,T by POLYRED:def 8;
    f reduces_to g,P,T by A8,POLYRED:def 7;
    hence f is_reducible_wrt P,T by POLYRED:def 9;
  end;
  then for f being non-zero Polynomial of n,L st f in I holds f
  is_top_reducible_wrt P,T by A1,Th26;
  then
  for b being bag of n st b in HT(I,T) ex b9 being bag of n st b9 in HT(P
  ,T) & b9 divides b by Th27;
  then HT(I,T) c= multiples(HT(P,T)) by Th28;
  hence thesis by A1,Th29;
end;
