
theorem Th1:
  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,
  p,g being Polynomial of n,L holds f reduces_to g,p,T implies ex m being
  Monomial of n,L st g = f - m *' p
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,p,g be
  Polynomial of n,L;
  assume f reduces_to g,p,T;
  then consider b being bag of n such that
A1: f reduces_to g,p,b,T by POLYRED:def 6;
  consider s being bag of n such that
  s + HT(p,T) = b and
A2: g = f - (f.b/HC(p,T)) * (s *' p) by A1,POLYRED:def 5;
  (f.b/HC(p,T)) * (s *' p) = Monom(f.b/HC(p,T),s) *' p by POLYRED:22;
  hence thesis by A2;
end;
