
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, G being Subset of Polynom-Ring(n,L)
  holds (for g1,g2,h being Polynomial of n,L st g1 in G & g2 in G & h
is_a_normal_form_of S-Poly(g1,g2,T),PolyRedRel(G,T) holds h = 0_(n,L)) implies
  (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 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, G be Subset of Polynom-Ring(n,L);
  set R = PolyRedRel(G,T);
  assume
A1: for g1,g2,h being Polynomial of n,L st g1 in G & g2 in G & h
  is_a_normal_form_of S-Poly(g1,g2,T),PolyRedRel(G,T) holds h = 0_(n,L);
  now
    let g1,g2 be Polynomial of n,L;
    now
      per cases;
      case
        S-Poly(g1,g2,T) in field R;
        hence S-Poly(g1,g2,T) has_a_normal_form_wrt R by REWRITE1:def 14;
      end;
      case
        not S-Poly(g1,g2,T) in field R;
        hence S-Poly(g1,g2,T) has_a_normal_form_wrt R by REWRITE1:46;
      end;
    end;
    then consider q being object such that
A2: q is_a_normal_form_of S-Poly(g1,g2,T),R by REWRITE1:def 11;
    R reduces S-Poly(g1,g2,T),q by A2,REWRITE1:def 6;
    then reconsider q as Polynomial of n,L by Lm1;
    assume g1 in G & g2 in G;
    then q = 0_(n,L) by A1,A2;
    hence R reduces S-Poly(g1,g2,T),0_(n,L) by A2,REWRITE1:def 6;
  end;
  hence thesis;
end;
