
theorem Th25: :: theorem 5.38 (i) ==> (ii), p. 207
  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, G,
  I being Subset of Polynom-Ring(n,L) holds (for f being Polynomial of n,L st f
  in I holds PolyRedRel(G,T) reduces f,0_(n,L)) implies (for f being non-zero
  Polynomial of n,L st f in I holds f is_reducible_wrt G,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, G,I be Subset
  of Polynom-Ring(n,L);
  assume
A1: for f being Polynomial of n,L st f in I holds PolyRedRel(G,T)
  reduces f,0_(n,L);
  now
    let f be non-zero Polynomial of n,L;
    assume f in I;
    then
A2: PolyRedRel(G,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,G,T & PolyRedRel(G,T)
    reduces g,0_(n,L) by A2,Lm5;
    hence f is_reducible_wrt G,T by POLYRED:def 9;
  end;
  hence thesis;
end;
