
theorem Th54:
  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) = Red(p1,T) *' p2 - Red(p2,T) *' p1
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;
  reconsider r1 = -Red(p1,T), r2 = -Red(p2,T) as Polynomial of n,L;
  r2 *' Red(p1,T) = -(Red(p2,T) *' Red(p1,T)) by POLYRED:6
    .= r1 *' Red(p2,T) by POLYRED:6;
  then
A1: (r2 *' Red(p1,T) + -(r1 *' Red(p2,T))) = 0_(n,L) by POLYRED:3;
  assume HT(p1,T),HT(p2,T) are_disjoint;
  hence S-Poly(p1,p2,T) = HM(p2,T) *' Red(p1,T) - (HM(p1,T) *' Red(p2,T)) by
Th53
    .= (p2 - Red(p2,T)) *' Red(p1,T) - (HM(p1,T) *' Red(p2,T)) by Th15
    .= (p2 - Red(p2,T)) *' Red(p1,T) - ((p1 - Red(p1,T)) *' Red(p2,T)) by Th15
    .= (p2 + -Red(p2,T)) *' Red(p1,T) - ((p1 - Red(p1,T)) *' Red(p2,T)) by
POLYNOM1:def 7
    .= (p2 + -Red(p2,T)) *' Red(p1,T) - ((p1 + -Red(p1,T)) *' Red(p2,T)) by
POLYNOM1:def 7
    .= (p2 *' Red(p1,T) + r2 *' Red(p1,T)) - ((p1 + -Red(p1,T)) *' Red(p2,T)
  ) by POLYNOM1:26
    .= (p2 *' Red(p1,T) + r2 *' Red(p1,T)) - (p1 *' Red(p2,T) + r1 *' Red(p2
  ,T)) by POLYNOM1:26
    .= (p2 *' Red(p1,T) + r2 *' Red(p1,T)) + -(p1 *' Red(p2,T) + r1 *' Red(
  p2,T)) by POLYNOM1:def 7
    .= (p2 *' Red(p1,T) + r2 *' Red(p1,T)) + (-(p1 *' Red(p2,T)) + -(r1 *'
  Red(p2,T))) by POLYRED:1
    .= p2 *' Red(p1,T) + (r2 *' Red(p1,T) + (-(r1 *' Red(p2,T)) + -(p1 *'
  Red(p2,T)))) by POLYNOM1:21
    .= p2 *' Red(p1,T) + (0_(n,L) + -(p1 *' Red(p2,T))) by A1,POLYNOM1:21
    .= p2 *' Red(p1,T) + -(p1 *' Red(p2,T)) by POLYRED:2
    .= Red(p1,T) *' p2 - Red(p2,T) *' p1 by POLYNOM1:def 7;
end;
