
theorem :: lemma 5.20 (ii), p. 197
  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 being Polynomial of n,L, m being non-zero Monomial of
  n,L holds f reduces_to f-m*'p,p,T implies HT(m*'p,T) in Support f
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 be Polynomial of n,L, m be non-zero Monomial of n,L;
  assume f reduces_to f-m*'p,p,T;
  then consider b being bag of n such that
A1: f reduces_to f-m*'p,p,b,T;
A2: p <> 0_(n,L) by A1;
  then
A3: p is non-zero;
A4: HC(p,T) <> 0.L by A2,TERMORD:17;
A5: now
    assume (HC(p,T))" = 0.L;
    then 0.L = HC(p,T) * (HC(p,T))"
      .= 1.L by A4,VECTSP_1:def 10;
    hence contradiction;
  end;
  b in Support f by A1;
  then f.b <> 0.L by POLYNOM1:def 4;
  then f.b * (HC(p,T))" <> 0.L by A5,VECTSP_2:def 1;
  then (f.b)/HC(p,T) <> 0.L;
  then
A6: (f.b)/HC(p,T) is non zero;
  consider s being bag of n such that
A7: s + HT(p,T) = b and
A8: f-m*'p = f-(((f.b)/HC(p,T)) * (s *' p)) by A1;
A9: (f.b)/HC(p,T) * (s *' p) = -- ((f.b)/HC(p,T) * (s *' p)) by POLYNOM1:19;
  f = f + 0_(n,L) by POLYNOM1:23
    .= f + (m*'p - m*' p) by POLYNOM1:24
    .= f + (m*'p + -(m*'p)) by POLYNOM1:def 7
    .= (f + -(m*'p)) + m*'p by POLYNOM1:21
    .= m*'p + (f -(f.b)/HC(p,T) * (s *' p)) by A8,POLYNOM1:def 7;
  then 0_(n,L) = f - (m*'p + (f -(f.b)/HC(p,T) * (s *' p))) by POLYNOM1:24
    .= f + -((m*'p) + (f -(f.b)/HC(p,T) * (s *' p))) by POLYNOM1:def 7
    .= f + (-(m*'p) + -(f -(f.b)/HC(p,T) * (s *' p))) by Th1
    .= f + (-(m*'p) + -(f + -(f.b)/HC(p,T) * (s *' p))) by POLYNOM1:def 7
    .= f + (-(m*'p) + (-f + -(-((f.b)/HC(p,T) * (s *' p))))) by Th1
    .= f + (-f + (-(m*'p) + (f.b)/HC(p,T) * (s *' p))) by A9,POLYNOM1:21
    .= (f + -f) + (-(m*'p) + (f.b)/HC(p,T) * (s *' p)) by POLYNOM1:21
    .= (f - f) + (-(m*'p) + (f.b)/HC(p,T) * (s *' p)) by POLYNOM1:def 7
    .= 0_(n,L) + (-(m*'p) + (f.b)/HC(p,T) * (s *' p)) by POLYNOM1:24
    .= -(m*'p) + (f.b)/HC(p,T) * (s *' p) by Th2;
  then m*'p = m*'p + (- m*'p + (f.b)/HC(p,T) * (s *' p)) by POLYNOM1:23
    .= (m*'p + - m*'p) + (f.b)/HC(p,T) * (s *' p) by POLYNOM1:21
    .= (m*'p - m*'p) + (f.b)/HC(p,T) * (s *' p) by POLYNOM1:def 7
    .= 0_(n,L) + (f.b)/HC(p,T) * (s *' p) by POLYNOM1:24
    .= (f.b)/HC(p,T) * (s *' p) by Th2;
  then HT(m*'p,T) = HT(s*'p,T) by A6,Th21
    .= b by A7,A3,Th15;
  hence thesis by A1;
end;
