
theorem Th15: :: theorem 5.35 (iv) ==> (v), p. 206
  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 empty doubleLoopStr, P being non empty Subset of
Polynom-Ring(n,L) holds PolyRedRel(P,T) is with_Church-Rosser_property implies
(for f being Polynomial of n,L st f in P-Ideal holds PolyRedRel(P,T) reduces f,
  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, P be non empty Subset of Polynom-Ring(n,L);
  set R = PolyRedRel(P,T);
  assume
A1: PolyRedRel(P,T) is with_Church-Rosser_property;
  now
    reconsider e = 0_(n,L) as Element of Polynom-Ring(n,L) by POLYNOM1:def 11;
    let f be Polynomial of n,L;
    assume
A2: f in P-Ideal;
    reconsider e as Element of Polynom-Ring(n,L);
    reconsider f9 = f as Element of Polynom-Ring(n,L) by POLYNOM1:def 11;
    reconsider f9 as Element of Polynom-Ring(n,L);
    f - 0_(n,L) = f9 - e by Lm2;
    then f9 - e in P-Ideal by A2,POLYRED:4;
    then f9,e are_congruent_mod P-Ideal by POLYRED:def 14;
    then f9,e are_convertible_wrt R by POLYRED:58;
    then f9,e are_convergent_wrt R by A1,REWRITE1:def 23;
    then consider c being object such that
A3: R reduces f,c and
A4: R reduces 0_(n,L),c by REWRITE1:def 7;
    reconsider c9 = c as Polynomial of n,L by A3,Lm4;
    now
      assume c9 <> 0_(n,L);
      then consider h being Polynomial of n,L such that
A5:   0_(n,L) reduces_to h,P,T and
      PolyRedRel(P,T) reduces h,c9 by A4,Lm5;
      consider pp being Polynomial of n,L such that
      pp in P and
A6:   0_(n,L) reduces_to h,pp,T by A5,POLYRED:def 7;
      0_(n,L) is_reducible_wrt pp,T by A6,POLYRED:def 8;
      hence contradiction by POLYRED:37;
    end;
    hence R reduces f,0_(n,L) by A3;
  end;
  hence thesis;
end;
