
theorem
  for n being Element of NAT, T being admissible connected TermOrder of
  n, L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
  degenerated non empty doubleLoopStr, G being Subset of Polynom-Ring(n,L) st
not(0_(n,L) in G) & for g being Polynomial of n,L st g in G holds g is Monomial
  of n,L holds G is_Groebner_basis_wrt T
proof
  let n being Element of NAT, T being admissible connected TermOrder of n, L
being add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non degenerated non
  empty doubleLoopStr, G being Subset of Polynom-Ring(n,L);
  assume that
A1: not 0_(n,L) in G and
A2: for g being Polynomial of n,L st g in G holds g is Monomial of n,L;
  now
    let g1,g2 be Polynomial of n,L;
    assume g1 in G & g2 in G;
    then g1 is Monomial of n,L & g2 is Monomial of n,L by A2;
    then S-Poly(g1,g2,T) = 0_(n,L) by Th19;
    hence PolyRedRel(G,T) reduces S-Poly(g1,g2,T),0_(n,L) by REWRITE1:12;
  end;
  hence thesis by A1,Th25;
end;
