
theorem Th20:
  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, p
being Polynomial of n,L holds S-Poly(p,0_(n,L),T) = 0_(n,L) & S-Poly(0_(n,L),p,
  T) = 0_(n,L)
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, p be
  Polynomial of n,L;
  set p2 = 0_(n,L);
  thus S-Poly(p,0_(n,L),T) = HC(0_(n,L),T) * (lcm(HT(p,T),HT(p2,T))/HT(p,T))
  *' p - Monom(HC(p,T),lcm(HT(p,T),HT(p2,T))/HT(p2,T)) *' 0_(n,L) by POLYRED:22
    .= HC(0_(n,L),T) * (lcm(HT(p,T),HT(p2,T))/HT(p,T)) *' p - 0_(n,L) by
POLYNOM1:28
    .= 0.L * ((lcm(HT(p,T),HT(p2,T))/HT(p,T)) *' p) - 0_(n,L) by TERMORD:17
    .= ((0.L |(n,L)) *' ((lcm(HT(p,T),HT(p2,T))/HT(p,T)) *' p)) - 0_(n,L) by
POLYNOM7:27
    .= (0_(n,L) *' ((lcm(HT(p,T),HT(p2,T))/HT(p,T)) *' p)) - 0_(n,L) by
POLYNOM7:19
    .= 0_(n,L) - 0_(n,L) by POLYRED:5
    .= 0_(n,L) by POLYRED:4;
  thus S-Poly(0_(n,L),p,T) = Monom(HC(p,T),(lcm(HT(p2,T),HT(p,T))/HT(p2,T)))
*' 0_(n,L) - HC(0_(n,L),T) * (lcm(HT(p2,T),HT(p,T))/HT(p,T)) *' p by POLYRED:22
    .= 0_(n,L) - HC(0_(n,L),T) * (lcm(HT(p2,T),HT(p,T))/HT(p,T)) *' p by
POLYNOM1:28
    .= 0_(n,L) - 0.L * ((lcm(HT(p2,T),HT(p,T))/HT(p,T)) *' p) by TERMORD:17
    .= 0_(n,L) - ((0.L |(n,L)) *' ((lcm(HT(p2,T),HT(p,T))/HT(p,T)) *' p)) by
POLYNOM7:27
    .= 0_(n,L) - (0_(n,L) *' ((lcm(HT(p2,T),HT(p,T))/HT(p,T)) *' p)) by
POLYNOM7:19
    .= 0_(n,L) - 0_(n,L) by POLYRED:5
    .= 0_(n,L) by POLYRED:4;
end;
