
theorem Th16: :: theorem 5.35 (v) ==> (vi), 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 Polynomial of n,L st f in
P-Ideal holds PolyRedRel(P,T) reduces f,0_(n,L)) implies (for f being non-zero
  Polynomial of n,L st f in P-Ideal holds f is_reducible_wrt P,T)
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 Polynomial of n,L st f in P-Ideal holds PolyRedRel(P,T)
  reduces f,0_(n,L);
  now
    let f be non-zero Polynomial of n,L;
    assume f in P-Ideal;
    then
A2: PolyRedRel(P,T) reduces f,0_(n,L) by A1;
    f <> 0_(n,L) by POLYNOM7:def 1;
    then ex g being Polynomial of n,L st f reduces_to g,P,T & PolyRedRel(P,T)
    reduces g,0_(n,L) by A2,Lm5;
    hence f is_reducible_wrt P,T by POLYRED:def 9;
  end;
  hence thesis;
end;
