
theorem Th41:
  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 f.b9 = g.b9
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: now
    assume b9 in Support(s*'p);
    then
A5: b9 <= b,T by A2,Th16;
    b <= b9,T by A1,TERMORD:def 3;
    then b = b9 by A5,TERMORD:7;
    hence contradiction by A1,TERMORD:def 3;
  end;
A6: b9 is Element of Bags n by PRE_POLY:def 12;
A7: ((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 A6,A4,POLYNOM1:def 4
    .= 0.L;
  (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 A7,POLYNOM1:17
    .= f.b9 + 0.L by RLVECT_1:12
    .= f.b9 by RLVECT_1:def 4;
  hence thesis by A3;
end;
