
theorem Th6:
  for n being Ordinal, T being connected TermOrder of n, L being
right_zeroed add-associative right_complementable well-unital distributive non
  trivial non empty doubleLoopStr, p being Polynomial of n,L holds HT(-p,T) =
  HT(p,T)
proof
  let n be Ordinal, T be connected TermOrder of n, L be right_zeroed
add-associative right_complementable well-unital distributive non trivial non
  empty doubleLoopStr, p be Polynomial of n,L;
  per cases;
  suppose
A1: p = 0_(n,L);
    reconsider x = -p as Element of Polynom-Ring(n,L) by POLYNOM1:def 11;
    reconsider x as Element of Polynom-Ring(n,L);
A2: -(0_(n,L)) = -(0_(n,L)) + 0_(n,L) by POLYNOM1:23
      .= 0_(n,L) by POLYRED:3;
    0.(Polynom-Ring(n,L)) = 0_(n,L) by POLYNOM1:def 11;
    then x + 0.(Polynom-Ring(n,L)) = (-p) + 0_(n,L) by POLYNOM1:def 11
      .= 0_(n,L) by A1,A2,POLYNOM1:23
      .= 0.(Polynom-Ring(n,L)) by POLYNOM1:def 11;
    then -p = -(0.Polynom-Ring(n,L)) by RLVECT_1:6
      .= 0.(Polynom-Ring(n,L)) by RLVECT_1:12
      .= p by A1,POLYNOM1:def 11;
    hence thesis;
  end;
  suppose
    p <> 0_(n,L);
    then
A3: Support(p) <> {} by POLYNOM7:1;
    then Support(-p) <> {} by Th5;
    then HT(-p,T) in Support(-p) by TERMORD:def 6;
    then HT(-p,T) in Support(p) by Th5;
    then
A4: HT(-p,T) <= HT(p,T),T by TERMORD:def 6;
    HT(p,T) in Support(p) by A3,TERMORD:def 6;
    then HT(p,T) in Support(-p) by Th5;
    then HT(p,T) <= HT(-p,T),T by TERMORD:def 6;
    hence thesis by A4,TERMORD:7;
  end;
end;
