
theorem
  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
degenerated non trivial doubleLoopStr, G being Subset of Polynom-Ring(n,L) st
not(0_(n,L) in G) holds (for g1,g2 being Polynomial of n,L st g1 in G & g2 in G
& not(HT(g1,T),HT(g2,T) are_disjoint) holds PolyRedRel(G,T) reduces S-Poly(g1,
g2,T),0_(n,L)) implies (for g1,g2,h being Polynomial of n,L st g1 in G & g2 in
G & not(HT(g1,T),HT(g2,T) are_disjoint) & h is_a_normal_form_of S-Poly(g1,g2,T)
  ,PolyRedRel(G,T) holds h = 0_(n,L))
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 degenerated non
  trivial doubleLoopStr, G be Subset of Polynom-Ring(n,L);
  assume
A1: not 0_(n,L) in G;
  assume
A2: for g1,g2 being Polynomial of n,L st g1 in G & g2 in G & not HT(g1,T
),HT(g2,T) are_disjoint holds PolyRedRel(G,T) reduces S-Poly(g1,g2,T),0_(n,L);
  for g1,g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel(
  G,T) reduces S-Poly(g1,g2,T),0_(n,L)
  proof
    let g1,g2 being Polynomial of n,L;
    assume that
A3: g1 in G and
A4: g2 in G;
    now
      per cases;
      case
A5:     HT(g1,T),HT(g2,T) are_disjoint;
        now
          let u be object;
          assume
A6:       u in {g1,g2};
          now
            per cases by A6,TARSKI:def 2;
            case
              u = g1;
              hence u in G by A3;
            end;
            case
              u = g2;
              hence u in G by A4;
            end;
          end;
          hence u in G;
        end;
        then
A7:     {g1,g2} c= G;
        PolyRedRel({g1,g2},T) reduces S-Poly(g1,g2,T),0_(n,L) by A5,Th56;
        hence thesis by A7,GROEB_1:4,REWRITE1:22;
      end;
      case
        not HT(g1,T),HT(g2,T) are_disjoint;
        hence thesis by A2,A3,A4;
      end;
    end;
    hence thesis;
  end;
  then G is_Groebner_basis_wrt T by A1,GROEB_2:25;
  hence thesis by GROEB_2:23;
end;
