
theorem
  for n being Ordinal, T being admissible connected 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, 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 & not(HT(m*'p,T) in
  Support g) & HT(m*'p,T) <= HT(f,T),T
proof
  let n be Ordinal, T be admissible connected TermOrder of n, L be
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non degenerated non
  empty 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;
  b in Support f by A1,POLYRED:def 5;
  then
A2: f.b <> 0.L by POLYNOM1:def 4;
 p <> 0_(n,L) by A1,POLYRED:def 5;
  then reconsider p as non-zero Polynomial of n,L by POLYNOM7:def 1;
  consider s being bag of n such that
A3: s + HT(p,T) = b and
A4: g = f - (f.b/HC(p,T)) * (s *' p) by A1,POLYRED:def 5;
  set m = Monom(f.b/HC(p,T),s);
A5: HC(p,T)" <> 0.L by VECTSP_1:25;
A6: f.b/HC(p,T) <> 0.L by A2,A5,VECTSP_2:def 1;
  then
A7: f.b/HC(p,T) is non zero;
  coefficient(m) <> 0.L by A6,POLYNOM7:9;
  then HC(m,T) <> 0.L by TERMORD:23;
  then m <> 0_(n,L) by TERMORD:17;
  then reconsider m as non-zero Monomial of n,L by POLYNOM7:def 1;
A8: HT(m*'p,T) = HT(m,T) + HT(p,T) by TERMORD:31
    .= term(m) + HT(p,T) by TERMORD:23
    .= s + HT(p,T) by A7,POLYNOM7:10;
  then HT(m*'p,T) in Support f by A1,A3,POLYRED:def 5;
  then
  (f.b/HC(p,T)) * (s *' p) = Monom(f.b/HC(p,T),s) *' p & HT(m*'p,T) <= HT
  (f,T) ,T by POLYRED:22,TERMORD:def 6;
  hence thesis by A1,A3,A4,A8,POLYRED:39;
end;
