
theorem
  for n being Ordinal, T being admissible 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 Polynomial of n,L holds S-Poly(p1,p2,T) = 0_(n,L) or HT(S-Poly(p1,p2,
  T),T) < lcm(HT(p1,T),HT(p2,T)),T
proof
  let n be Ordinal, T be admissible 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 Polynomial of n,L;
  assume
A1: S-Poly(p1,p2,T) <> 0_(n,L);
  set sp = S-Poly(p1,p2,T), g1 = HC(p2,T) * (lcm(HT(p1,T),HT(p2,T))/HT(p1,T))
  *' p1, g2 = HC(p1,T) * (lcm(HT(p1,T),HT(p2,T))/HT(p2,T)) *' p2;
  per cases;
  suppose
    p1 = 0_(n,L) or p2 = 0_(n,L);
    hence thesis by A1,Th20;
  end;
  suppose
A2: p1 <> 0_(n,L) & p2 <> 0_(n,L);
    then
A3: HC(p2,T) <> 0.L by TERMORD:17;
    then
A4: HC(p2,T) is non zero;
A5: HT(Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T))),T) = term(Monom(
    HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))) by TERMORD:23
      .= lcm(HT(p1,T),HT(p2,T))/HT(p1,T) by A4,POLYNOM7:10;
A6: p1 is non-zero by A2,POLYNOM7:def 1;
    HC(Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T))),T) = coefficient
    (Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))) by TERMORD:23
      .= HC(p2,T) by POLYNOM7:9;
    then Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T))) <> 0_(n,L) by A3,
TERMORD:17;
    then
A7: Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T))) is non-zero by
POLYNOM7:def 1;
A8: HC(g1,T) = HC((Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))) *'
    p1,T) by POLYRED:22
      .= HC((Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))),T) * HC(p1,T
    ) by A6,A7,TERMORD:32
      .= coefficient(Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))) * HC
    (p1,T) by TERMORD:23
      .= HC(p1,T) * HC(p2,T) by POLYNOM7:9;
A9: HT(p1,T) divides lcm(HT(p1,T),HT(p2,T)) by Th3;
A10: HT(g1,T) = HT((Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))) *'
    p1,T) by POLYRED:22
      .= HT((Monom(HC(p2,T),(lcm(HT(p1,T),HT(p2,T))/HT(p1,T)))),T) + HT(p1,T
    ) by A6,A7,TERMORD:31
      .= lcm(HT(p1,T),HT(p2,T)) by A9,A5,Def1;
A11: HC(p1,T) <> 0.L by A2,TERMORD:17;
    then
A12: HC(p1,T) is non zero;
A13: HT(Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T))),T) = term(Monom(
    HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))) by TERMORD:23
      .= lcm(HT(p1,T),HT(p2,T))/HT(p2,T) by A12,POLYNOM7:10;
A14: p2 is non-zero by A2,POLYNOM7:def 1;
    HC(Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T))),T) = coefficient
    (Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))) by TERMORD:23
      .= HC(p1,T) by POLYNOM7:9;
    then Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T))) <> 0_(n,L) by A11,
TERMORD:17;
    then
A15: Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T))) is non-zero by
POLYNOM7:def 1;
    Support sp <> {} by A1,POLYNOM7:1;
    then
A16: HT(sp,T) in Support sp by TERMORD:def 6;
A17: HT(p2,T) divides lcm(HT(p1,T),HT(p2,T)) by Th3;
A18: HC(g2,T) = HC((Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))) *'
    p2,T) by POLYRED:22
      .= HC((Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))),T) * HC(p2,T
    ) by A14,A15,TERMORD:32
      .= coefficient(Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))) * HC
    (p2,T) by TERMORD:23
      .= HC(p1,T) * HC(p2,T) by POLYNOM7:9;
A19: HT(g2,T) = HT((Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))) *'
    p2,T) by POLYRED:22
      .= HT((Monom(HC(p1,T),(lcm(HT(p1,T),HT(p2,T))/HT(p2,T)))),T) + HT(p2,T
    ) by A14,A15,TERMORD:31
      .= lcm(HT(p1,T),HT(p2,T)) by A17,A13,Def1;
    then sp.(lcm(HT(p1,T),HT(p2,T))) = (g1+-g2).HT(g2,T) by POLYNOM1:def 7
      .= g1.HT(g2,T) + (-g2).HT(g2,T) by POLYNOM1:15
      .= g1.HT(g2,T) + -(g2.HT(g2,T)) by POLYNOM1:17
      .= HC(g1,T) + -(g2.HT(g2,T)) by A10,A19,TERMORD:def 7
      .= HC(g1,T) + -HC(g2,T) by TERMORD:def 7
      .= 0.L by A8,A18,RLVECT_1:5;
    then
A20: not lcm(HT(p1,T),HT(p2,T)) in Support sp by POLYNOM1:def 4;
    HT(sp,T) <= max(HT(g1,T),HT(g2,T),T), T by GROEB_1:7;
    then HT(sp,T) <= lcm(HT(p1,T),HT(p2,T)),T by A10,A19,TERMORD:12;
    hence thesis by A16,A20,TERMORD:def 3;
  end;
end;
