
theorem :: lemma 5.20 (i), p. 197
  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, f
  being Polynomial of n,L, p being non-zero Polynomial of n,L holds f
  is_reducible_wrt p,T iff ex b being bag of n st b in Support f & HT(p,T)
  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, f be
  Polynomial of n,L, p be non-zero Polynomial of n,L;
A1: now
A2: p <> 0_(n,L) by POLYNOM7:def 1;
    assume ex b being bag of n st b in Support f & HT(p,T) divides b;
    then consider b being bag of n such that
A3: b in Support f and
A4: HT(p,T) divides b;
    consider s being bag of n such that
A5: b = HT(p,T) + s by A4,TERMORD:1;
    set g = f - ((f.b)/HC(p,T)) * (s *' p);
    f <> 0_(n,L) by A3,POLYNOM7:1;
    then f reduces_to g,p,b,T by A3,A5,A2;
    then f reduces_to g,p,T;
    hence f is_reducible_wrt p,T;
  end;
  now
    assume f is_reducible_wrt p,T;
    then consider g being Polynomial of n,L such that
A6: f reduces_to g,p,T;
    consider b being bag of n such that
A7: f reduces_to g,p,b,T by A6;
    ex s being bag of n st s + HT(p,T) = b & g = f - (f.b)/HC(p,T) * (s *'
    p ) by A7;
    then
A8: HT(p,T) divides b by TERMORD:1;
    b in Support f by A7;
    hence ex b being bag of n st b in Support f & HT(p,T) divides b by A8;
  end;
  hence thesis by A1;
end;
