
theorem Th18: :: theorem 5.35 (vii) ==> (viii), p. 206
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, P
being Subset of Polynom-Ring(n,L) holds (for f being non-zero Polynomial of n,L
st f in P-Ideal holds f is_top_reducible_wrt P,T) implies (for b being bag of n
  st b in HT(P-Ideal,T) ex b9 being bag of n st b9 in HT(P,T) & b9 divides b)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_complementable right_zeroed commutative associative well-unital
distributive almost_left_invertible non trivial doubleLoopStr, P be Subset of
  Polynom-Ring(n,L);
  assume
A1: for f being non-zero Polynomial of n,L st f in P-Ideal holds f
  is_top_reducible_wrt P,T;
  now
    let b be bag of n;
    assume b in HT(P-Ideal,T);
    then consider p being Polynomial of n,L such that
A2: b = HT(p,T) and
A3: p in P-Ideal and
A4: p <> 0_(n,L);
    reconsider p as non-zero Polynomial of n,L by A4,POLYNOM7:def 1;
    p is_top_reducible_wrt P,T by A1,A3;
    then consider u being Polynomial of n,L such that
A5: u in P and
A6: p is_top_reducible_wrt u,T by POLYRED:def 12;
    consider q being Polynomial of n,L such that
A7: p top_reduces_to q,u,T by A6,POLYRED:def 11;
A8: p reduces_to q,u,HT(p,T),T by A7,POLYRED:def 10;
    then u <> 0_(n,L) by POLYRED:def 5;
    then
A9: HT(u,T) in {HT(r,T) where r is Polynomial of n,L : r in P & r <> 0_(n
    ,L)} by A5;
    ex s being bag of n st s + HT(u,T) = HT(p,T) & q = p - ( p.(HT(p,T))/
    HC(u,T)) * (s *' u) by A8,POLYRED:def 5;
    hence ex b9 being bag of n st b9 in HT(P,T) & b9 divides b by A2,A9,
PRE_POLY:50;
  end;
  hence thesis;
end;
