
theorem Th19:
  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, m1
  ,m2 being Monomial of n,L holds S-Poly(m1,m2,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, m1,m2 be
  Monomial of n,L;
  per cases;
  suppose
A1: m1 = 0_(n,L);
A2: HC(Monom(HC(0_(n,L),T),(lcm(HT(m1,T),HT(m2,T))/HT(m2,T))),T) =
coefficient(Monom(HC(0_(n,L),T),(lcm(HT(m1,T),HT(m2,T))/HT(m2,T)))) by
TERMORD:23
      .= HC(0_(n,L),T) by POLYNOM7:9
      .= 0.L by TERMORD:17;
    thus S-Poly(m1,m2,T) = Monom(HC(m2,T),(lcm(HT(m1,T),HT(m2,T))/HT(m1,T)))
    *' 0_(n,L) - HC(0_(n,L),T) * (lcm(HT(m1,T),HT(m2,T))/HT(m2,T)) *' m2 by A1,
POLYRED:22
      .= 0_(n,L) - HC(0_(n,L),T) * (lcm(HT(m1,T),HT(m2,T))/HT(m2,T)) *' m2
    by POLYNOM1:28
      .= 0_(n,L) - Monom(HC(0_(n,L),T),(lcm(HT(m1,T),HT(m2,T))/HT(m2,T))) *'
    m2 by POLYRED:22
      .= 0_(n,L) - 0_(n,L) *' m2 by A2,TERMORD:17
      .= 0_(n,L) - 0_(n,L) by POLYRED:5
      .= 0_(n,L) by POLYNOM1:24;
  end;
  suppose
A3: m2 = 0_(n,L);
A4: HC(Monom(HC(0_(n,L),T),(lcm(HT(m1,T),HT(m2,T))/HT(m1,T))),T) =
coefficient(Monom(HC(0_(n,L),T),(lcm(HT(m1,T),HT(m2,T))/HT(m1,T)))) by
TERMORD:23
      .= HC(0_(n,L),T) by POLYNOM7:9
      .= 0.L by TERMORD:17;
    thus S-Poly(m1,m2,T) = HC(0_(n,L),T) * (lcm(HT(m1,T),HT(m2,T))/HT(m1,T))
    *' m1 - Monom(HC(m1,T),(lcm(HT(m1,T),HT(m2,T))/HT(m2,T))) *' 0_(n,L) by A3,
POLYRED:22
      .= HC(0_(n,L),T) * (lcm(HT(m1,T),HT(m2,T))/HT(m1,T)) *' m1 - 0_(n,L)
    by POLYNOM1:28
      .= Monom(HC(0_(n,L),T),(lcm(HT(m1,T),HT(m2,T))/HT(m1,T))) *' m1 - 0_(n
    ,L) by POLYRED:22
      .= 0_(n,L) *' m1 - 0_(n,L) by A4,TERMORD:17
      .= 0_(n,L) - 0_(n,L) by POLYRED:5
      .= 0_(n,L) by POLYNOM1:24;
  end;
  suppose
A5: m1 <> 0_(n,L) & m2 <> 0_(n,L);
    then HC(m2,T) <> 0.L by TERMORD:17;
    then
A6: HC(m2,T) is non zero;
    HC(m1,T) <> 0.L by A5,TERMORD:17;
    then
A7: HC(m1,T) is non zero;
A8: HT(m2,T) divides lcm(HT(m1,T),HT(m2,T)) by Th3;
A9: m2 = Monom(coefficient(m2),term(m2)) by POLYNOM7:11
      .= Monom(HC(m2,T),term(m2)) by TERMORD:23
      .= Monom(HC(m2,T),HT(m2,T)) by TERMORD:23;
A10: HT(m1,T) divides lcm(HT(m1,T),HT(m2,T)) by Th3;
A11: m1 = Monom(coefficient(m1),term(m1)) by POLYNOM7:11
      .= Monom(HC(m1,T),term(m1)) by TERMORD:23
      .= Monom(HC(m1,T),HT(m1,T)) by TERMORD:23;
A12: HC(m1,T) * (lcm(HT(m1,T),HT(m2,T))/HT(m2,T)) *' m2 = Monom(HC(m1,T),(
    lcm(HT(m1,T),HT(m2,T))/HT(m2,T))) *' m2 by POLYRED:22
      .= Monom(HC(m2,T)*HC(m1,T), (lcm(HT(m1,T),HT(m2,T))/HT(m2,T))+HT(m2,T)
    ) by A7,A6,A9,TERMORD:3
      .= Monom(HC(m2,T)*HC(m1,T), lcm(HT(m1,T),HT(m2,T))) by A8,Def1;
    HC(m2,T) * (lcm(HT(m1,T),HT(m2,T))/HT(m1,T)) *' m1 = Monom(HC(m2,T),(
    lcm(HT(m1,T),HT(m2,T))/HT(m1,T))) *' m1 by POLYRED:22
      .= Monom(HC(m2,T)*HC(m1,T), (lcm(HT(m1,T),HT(m2,T))/HT(m1,T))+HT(m1,T)
    ) by A7,A6,A11,TERMORD:3
      .= Monom(HC(m2,T)*HC(m1,T), lcm(HT(m1,T),HT(m2,T))) by A10,Def1;
    hence thesis by A12,POLYNOM1:24;
  end;
end;
