
theorem
  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 being non-zero Polynomial of n,L, p being Polynomial of n,L
  st p.(HT(f*'g,T)) = 0.L holds f*'g+p reduces_to Red(f,T)*'g+p,{g},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 be non-zero Polynomial of n,L, p be Polynomial of n,L;
  assume
A1: p.(HT(f*'g,T)) = 0.L;
  f*'g <> 0_(n,L) by POLYNOM7:def 1;
  then Support(f*'g) <> {} by POLYNOM7:1;
  then HT(f*'g,T) in Support(f*'g) by TERMORD:def 6;
  then
A2: (f*'g).HT(f*'g,T) <> 0.L by POLYNOM1:def 4;
  reconsider r = -HM(f,T) as Polynomial of n,L;
  set fg = f*'g+p;
  set q = fg - ((fg).HT(f*'g,T)/HC(g,T)) * (HT(f,T) *' g);
A3: HT(f*'g,T) = HT(f,T) + HT(g,T) by TERMORD:31;
A4: g <> 0_(n,L) by POLYNOM7:def 1;
A5: HC(g,T) <> 0.L;
  fg.(HT(f*'g,T)) = (f*'g).HT(f*'g,T) + p.HT(f*'g,T) by POLYNOM1:15
    .= (f*'g).HT(f*'g,T) by A1,RLVECT_1:def 4;
  then
A6: HT(f*'g,T) in Support fg by A2,POLYNOM1:def 4;
  then fg <> 0_(n,L) by POLYNOM7:1;
  then fg reduces_to q,g,HT(f*'g,T),T by A6,A4,A3,POLYRED:def 5;
  then
A7: g in {g} & fg reduces_to q,g,T by POLYRED:def 6,TARSKI:def 1;
  q = fg - (((f*'g).HT(f*'g,T)+0.L)/HC(g,T)) * (HT(f,T) *' g) by A1,POLYNOM1:15
    .= fg - ((f*'g).HT(f*'g,T)/HC(g,T)) * (HT(f,T) *' g) by RLVECT_1:def 4
    .= fg - (HC(f*'g,T)/HC(g,T)) * (HT(f,T) *' g) by TERMORD:def 7
    .= fg - ((HC(f,T)*HC(g,T))/HC(g,T)) * (HT(f,T) *' g) by TERMORD:32
    .= fg - ((HC(f,T)*HC(g,T))*HC(g,T)") * (HT(f,T) *' g)
    .= fg - (HC(f,T)*(HC(g,T)*HC(g,T)")) * (HT(f,T) *' g) by GROUP_1:def 3
    .= fg - (HC(f,T)*1.L) * (HT(f,T) *' g) by A5,VECTSP_1:def 10
    .= fg - HC(f,T) * (HT(f,T) *' g)
    .= fg - Monom(HC(f,T),HT(f,T)) *' g by POLYRED:22
    .= fg - HM(f,T) *' g by TERMORD:def 8
    .= fg + -(HM(f,T) *' g) by POLYNOM1:def 7
    .= (f*'g+p) + (r *' g) by POLYRED:6
    .= (f*'g + r *' g) + p by POLYNOM1:21
    .= g *' (f + -HM(f,T)) + p by POLYNOM1:26
    .= (f - HM(f,T)) *' g + p by POLYNOM1:def 7
    .= Red(f,T) *' g + p by TERMORD:def 9;
  hence thesis by A7,POLYRED:def 7;
end;
