
theorem Th45:
  for n being Ordinal, T being connected admissible TermOrder of n
  , L being add-associative right_complementable right_zeroed commutative
  associative well-unital distributive Abelian almost_left_invertible non
  trivial doubleLoopStr, f,g,p being Polynomial of n,L st f reduces_to g,p,T
  holds -f reduces_to -g,p,T
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 Abelian almost_left_invertible non trivial
  doubleLoopStr, f,g,p 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: b in Support -f by GROEB_1:5;
  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;
  g = f + (-((f.b/HC(p,T)) * (s *' p))) by A4,POLYNOM1:def 7;
  then
A5: -g = -f + -(-((f.b/HC(p,T)) * (s *' p))) by POLYRED:1
    .= -f - -((f.b/HC(p,T)) * (s *' p)) by POLYNOM1:def 7
    .= -f - (-(f.b/HC(p,T))) * (s *' p) by POLYRED:9
    .= -f - (-(f.b*(HC(p,T))")) * (s *' p)
    .= -f - ((-f.b)*(HC(p,T))") * (s *' p) by VECTSP_1:9
    .= -f - ((-f.b)/(HC(p,T))) * (s *' p)
    .= -f - ((-f).b/HC(p,T)) * (s *' p) by POLYNOM1:17;
A6: now
A7:   --f = f by POLYNOM1:19;
    assume -f = 0_(n,L);
    then f = -(0_(n,L)) by A7 .= 0_(n,L) by Th13;
    hence contradiction by A1,POLYRED:def 5;
  end;
  p <> 0_(n,L) by A1,POLYRED:def 5;
  then -f reduces_to -g,p,b,T by A3,A6,A5,A2,POLYRED:def 5;
  hence thesis by POLYRED:def 6;
end;
