
theorem Th21:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable distributive domRing-like
non trivial doubleLoopStr, p being Polynomial of n,L, a being non zero Element
  of L holds HT(a*p,T) = HT(p,T)
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
  right_zeroed right_complementable distributive domRing-like non trivial
  doubleLoopStr, p be Polynomial of n,L, a be non zero Element of L;
  set ht = HT(a*p,T), htp = HT(p,T);
  per cases;
  suppose
A1: Support(a*p) = {};
    now
      assume Support p <> {};
      then reconsider sp = Support p as non empty set;
      set u = the Element of sp;
      u in Support p;
      then reconsider u9 = u as Element of Bags n;
A2:   (a*p).u9 = a * p.u9 by POLYNOM7:def 9;
      p.u9 <> 0.L & a <> 0.L by POLYNOM1:def 4;
      then (a*p).u9 <> 0.L by A2,VECTSP_2:def 1;
      hence contradiction by A1,POLYNOM1:def 4;
    end;
    hence htp = EmptyBag n by TERMORD:def 6
      .= ht by A1,TERMORD:def 6;
  end;
  suppose
A3: Support(a*p) <> {};
    now
      reconsider sp = Support(a*p) as non empty set by A3;
      set u = the Element of sp;
      u in Support(a*p);
      then reconsider u9 = u as Element of Bags n;
      (a*p).u9 <> 0.L & (a*p).u9 = a * p.u9 by POLYNOM1:def 4,POLYNOM7:def 9;
      then
A4:   p.u9 <> 0.L;
      assume Support p = {};
      hence contradiction by A4,POLYNOM1:def 4;
    end;
    then htp in Support p by TERMORD:def 6;
    then
A5: p.htp <> 0.L by POLYNOM1:def 4;
A6: now
      let b be bag of n;
      assume
A7:   b in Support(a*p);
      Support(a*p) c= Support(p) by Th19;
      hence b <= htp,T by A7,TERMORD:def 6;
    end;
    (a*p).htp = a * p.htp by POLYNOM7:def 9;
    then (a*p).htp <> 0.L by A5,VECTSP_2:def 1;
    then htp in Support(a*p) by POLYNOM1:def 4;
    hence thesis by A6,TERMORD:def 6;
  end;
end;
