
theorem Th47:
  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 holds f*'g
  reduces_to Red(f,T)*'g,{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;
  set fg = f*'g;
  set q = fg - (fg.HT(fg,T)/HC(g,T)) * (HT(f,T) *' g);
  reconsider r = -HM(f,T) as Polynomial of n,L;
A1: g <> 0_(n,L) by POLYNOM7:def 1;
A2: HC(g,T) <> 0.L;
A3: fg <> 0_(n,L) by POLYNOM7:def 1;
  then Support fg <> {} by POLYNOM7:1;
  then
A4: HT(fg,T) in Support fg by TERMORD:def 6;
  HT(fg,T) = HT(f,T) + HT(g,T) by TERMORD:31;
  then fg reduces_to q,g,HT(fg,T),T by A3,A1,A4,POLYRED:def 5;
  then
A5: g in {g} & fg reduces_to q,g,T by POLYRED:def 6,TARSKI:def 1;
  q = fg - (HC(fg,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 A2,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
    .= fg + (r *' g) by POLYRED:6
    .= g *' (f + -HM(f,T)) by POLYNOM1:26
    .= (f - HM(f,T)) *' g by POLYNOM1:def 7
    .= Red(f,T) *' g by TERMORD:def 9;
  hence thesis by A5,POLYRED:def 7;
end;
