
theorem Th53:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_complementable right_zeroed commutative associative
  well-unital distributive Abelian almost_left_invertible non trivial
doubleLoopStr, p1,p2 being Polynomial of n,L st HT(p1,T),HT(p2,T) are_disjoint
  holds S-Poly(p1,p2,T) = HM(p2,T) *' Red(p1,T) - HM(p1,T) *' Red(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 Abelian almost_left_invertible non trivial doubleLoopStr, p1,p2
  be Polynomial of n,L;
  assume HT(p1,T),HT(p2,T) are_disjoint;
  then lcm(HT(p1,T),HT(p2,T)) = HT(p1,T) + HT(p2,T) by GROEB_2:5;
  hence S-Poly(p1,p2,T) = HC(p2,T) * ((HT(p1,T) + HT(p2,T))/HT(p1,T)) *' p1 -
  HC(p1,T) * ((HT(p1,T) + HT(p2,T))/HT(p2,T)) *' p2 by GROEB_2:def 4
    .= HC(p2,T) * HT(p2,T) *' p1 - HC(p1,T) * ((HT(p1,T) + HT(p2,T))/HT(p2,T
  )) *' p2 by Th1
    .= HC(p2,T) * HT(p2,T) *' p1 - HC(p1,T) * HT(p1,T) *' p2 by Th1
    .= HC(p2,T) * HT(p2,T) *' (HM(p1,T) + Red(p1,T)) - HC(p1,T) * HT(p1,T)
  *' p2 by TERMORD:38
    .= HC(p2,T) * HT(p2,T) *' (HM(p1,T) + Red(p1,T)) - HC(p1,T) * HT(p1,T)
  *' (HM(p2,T) + Red(p2,T)) by TERMORD:38
    .= Monom(HC(p2,T),HT(p2,T)) *' (HM(p1,T) + Red(p1,T)) - HC(p1,T) * HT(p1
  ,T) *' (HM(p2,T) + Red(p2,T)) by POLYRED:22
    .= Monom(HC(p2,T),HT(p2,T)) *' (HM(p1,T) + Red(p1,T)) - Monom(HC(p1,T),
  HT(p1,T)) *' (HM(p2,T) + Red(p2,T)) by POLYRED:22
    .= HM(p2,T) *' (HM(p1,T) + Red(p1,T)) - Monom(HC(p1,T),HT(p1,T)) *' (HM(
  p2,T) + Red(p2,T)) by TERMORD:def 8
    .= HM(p2,T) *' (HM(p1,T) + Red(p1,T)) - HM(p1,T) *' (HM(p2,T) + Red(p2,T
  )) by TERMORD:def 8
    .= (HM(p2,T) *' HM(p1,T) + HM(p2,T) *' Red(p1,T)) - HM(p1,T) *' (HM(p2,T
  ) + Red(p2,T)) by POLYNOM1:26
    .= (HM(p2,T) *' HM(p1,T) + HM(p2,T) *' Red(p1,T)) - (HM(p1,T) *' HM(p2,T
  ) + HM(p1,T) *' Red(p2,T)) by POLYNOM1:26
    .= (HM(p2,T) *' HM(p1,T) + HM(p2,T) *' Red(p1,T)) + -((HM(p1,T) *' HM(p2
  ,T)) + (HM(p1,T) *' Red(p2,T))) by POLYNOM1:def 7
    .= (HM(p2,T) *' HM(p1,T) + HM(p2,T) *' Red(p1,T)) + (-(HM(p1,T) *' HM(p2
  ,T)) + -(HM(p1,T) *' Red(p2,T))) by POLYRED:1
    .= HM(p2,T) *' Red(p1,T) + (HM(p2,T) *' HM(p1,T) + (-(HM(p1,T) *' HM(p2,
  T)) + -(HM(p1,T) *' Red(p2,T)))) by POLYNOM1:21
    .= HM(p2,T) *' Red(p1,T) + ((HM(p2,T) *' HM(p1,T) + -(HM(p1,T) *' HM(p2,
  T))) + -(HM(p1,T) *' Red(p2,T))) by POLYNOM1:21
    .= HM(p2,T) *' Red(p1,T) + (0_(n,L) + -(HM(p1,T) *' Red(p2,T))) by
POLYRED:3
    .= HM(p2,T) *' Red(p1,T) + -(HM(p1,T) *' Red(p2,T)) by POLYRED:2
    .= HM(p2,T) *' Red(p1,T) - (HM(p1,T) *' Red(p2,T)) by POLYNOM1:def 7;
end;
