
theorem
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
well-unital distributive almost_left_invertible non trivial doubleLoopStr, p1
,p2 being non-zero Polynomial of n,L holds HT(p2,T) divides HT(p1,T) implies HC
  (p2,T) * p1 top_reduces_to S-Poly(p1,p2,T),p2,T
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_complementable right_zeroed commutative associative well-unital
  distributive almost_left_invertible non trivial doubleLoopStr, p1,p2 be
  non-zero Polynomial of n,L;
  set hcp2 = HC(p2,T);
  assume
A1: HT(p2,T) divides HT(p1,T);
  then consider b being bag of n such that
A2: HT(p1,T) = HT(p2,T) + b by TERMORD:1;
  set g = (hcp2*p1) - ((hcp2*p1).HT(p1,T)/HC(p2,T)) * (b *' p2);
A3: p2 <> 0_(n,L) by POLYNOM7:def 1;
A4: hcp2 <> 0.L;
  p1 <> 0_(n,L) by POLYNOM7:def 1;
  then Support p1 <> {} by POLYNOM7:1;
  then
A5: HT(p1,T) in Support p1 by TERMORD:def 6;
A6: Support(p1) c= Support(hcp2*p1) by POLYRED:20;
  then hcp2*p1 <> 0_(n,L) by A5,POLYNOM7:1;
  then
A7: HT(hcp2*p1,T) = HT(p1,T) & hcp2*p1 reduces_to g,p2,HT(p1,T),T by A3,A5,A2
,A6,POLYRED:21,def 5;
A8: lcm(HT(p1,T),HT(p2,T)) = HT(p1,T) by A1,Th7;
  g = (hcp2*p1) - (hcp2*(p1.HT(p1,T))/HC(p2,T)) * (b *' p2) by POLYNOM7:def 9
    .= (hcp2*p1) - ((hcp2*HC(p1,T))/HC(p2,T)) * (b *' p2) by TERMORD:def 7
    .= (hcp2*p1) - ((hcp2*HC(p1,T))*(HC(p2,T)")) * (b *' p2)
    .= (hcp2*p1) - (HC(p1,T)*(hcp2*(HC(p2,T)"))) * (b *' p2) by GROUP_1:def 3
    .= (hcp2*p1) - (HC(p1,T)*1.L) * (b *' p2) by A4,VECTSP_1:def 10
    .= (hcp2*p1) - HC(p1,T) * (b *' p2)
    .= HC(p2,T) * (EmptyBag n) *' p1 - HC(p1,T) * (b *' p2) by POLYRED:17
    .= HC(p2,T) * (HT(p1,T)/HT(p1,T)) *' p1 - HC(p1,T) * (b *' p2) by Th6
    .= S-Poly(p1,p2,T) by A1,A2,A8,Def1;
  hence thesis by A7,POLYRED:def 10;
end;
