
theorem Th40: :: lemma 5.20 (iii), p. 197
  for n being Ordinal, T being connected admissible 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, b,b9 being bag of n st b < b9,T
  holds f reduces_to g,p,b,T implies (b9 in Support g iff b9 in Support f)
proof
  let n be Ordinal, T be connected admissible 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, b,b9 be bag of n;
  assume
A1: b < b9,T;
  assume f reduces_to g,p,b,T;
  then consider s being bag of n such that
A2: s + HT(p,T) = b and
A3: g = f - (f.b/HC(p,T)) * (s *' p);
A4: b9 is Element of Bags n by PRE_POLY:def 12;
A5: now
    assume b9 in Support(s*'p);
    then
A6: b9 <= b,T by A2,Th16;
    b <= b9,T by A1,TERMORD:def 3;
    then b = b9 by A6,TERMORD:7;
    hence contradiction by A1,TERMORD:def 3;
  end;
A7: now
A8: ((f.b/HC(p,T)) * (s *' p)).b9 = (f.b/HC(p,T)) * (s *' p).b9 by
POLYNOM7:def 9
      .= (f.b/HC(p,T)) * 0.L by A4,A5,POLYNOM1:def 4
      .= 0.L;
    assume
A9: b9 in Support f;
    (f - (f.b/HC(p,T)) * (s *' p)).b9 = (f + -((f.b/HC(p,T)) * (s *' p)))
    .b9 by POLYNOM1:def 7
      .= f.b9 + (-(f.b/HC(p,T) * (s *' p))).b9 by POLYNOM1:15
      .= f.b9 + -0.L by A8,POLYNOM1:17
      .= f.b9 + 0.L by RLVECT_1:12
      .= f.b9 by RLVECT_1:def 4;
    then g.b9 <> 0.L by A3,A9,POLYNOM1:def 4;
    hence b9 in Support g by A4,POLYNOM1:def 4;
  end;
  now
A10: ((f.b/HC(p,T)) * (s *' p)).b9 = (f.b/HC(p,T)) * (s *' p).b9 by
POLYNOM7:def 9
      .= (f.b/HC(p,T)) * 0.L by A4,A5,POLYNOM1:def 4
      .= 0.L;
    assume
A11: b9 in Support g;
    (f - (f.b/HC(p,T)) * (s *' p)).b9 = (f + -((f.b/HC(p,T)) * (s *' p)))
    .b9 by POLYNOM1:def 7
      .= f.b9 + (-(f.b/HC(p,T) * (s *' p))).b9 by POLYNOM1:15
      .= f.b9 + -0.L by A10,POLYNOM1:17
      .= f.b9 + 0.L by RLVECT_1:12
      .= f.b9 by RLVECT_1:def 4;
    then f.b9 <> 0.L by A3,A11,POLYNOM1:def 4;
    hence b9 in Support f by A4,POLYNOM1:def 4;
  end;
  hence thesis by A7;
end;
