
theorem Th15:
  for n being Ordinal, T being connected admissible TermOrder of n
, L being non trivial ZeroStr, p being non-zero Polynomial of n,L, b being bag
  of n holds HT(b*'p,T) = b + HT(p,T)
proof
  let n be Ordinal, T be connected admissible TermOrder of n, L be non trivial
  ZeroStr, p be non-zero Polynomial of n,L, b be bag of n;
  set htp = HT(p,T);
  per cases;
  suppose
A1: Support(b*'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;
      b divides b+u9 by PRE_POLY:50;
      then (b*'p).(b+u9) = p.(b+u9-'b) by Def1
        .= p.u9 by PRE_POLY:48;
      then
A2:   (b*'p).(b+u9) <> 0.L by POLYNOM1:def 4;
      b+u9 is Element of Bags n by PRE_POLY:def 12;
      hence contradiction by A1,A2,POLYNOM1:def 4;
    end;
    then p = 0_(n,L) by POLYNOM7:1;
    hence thesis by POLYNOM7:def 1;
  end;
  suppose
A3: Support(b*'p) <> {};
    now
      reconsider sp = Support(b*'p) as non empty set by A3;
      set u = the Element of sp;
      u in Support(b*'p);
      then reconsider u9 = u as Element of Bags n;
A4:   u9-'b is Element of Bags n by PRE_POLY:def 12;
A5:   (b*'p).u9 <> 0.L by POLYNOM1:def 4;
      then b divides u9 by Def1;
      then
A6:   p.(u9-'b) <> 0.L by A5,Def1;
      assume Support p = {};
      hence contradiction by A6,A4,POLYNOM1:def 4;
    end;
    then htp in Support p by TERMORD:def 6;
    then
A7: p.htp <> 0.L by POLYNOM1:def 4;
A8: now
      let b9 be bag of n;
      assume b9 in Support(b*'p);
      then
A9:   (b*'p).b9 <> 0.L by POLYNOM1:def 4;
      then b divides b9 by Def1;
      then consider b3 being bag of n such that
A10:  b + b3 = b9 by TERMORD:1;
A11:  b3 is Element of Bags n by PRE_POLY:def 12;
      (b*'p).b9 = p.b3 by A10,Lm9;
      then b3 in Support p by A9,A11,POLYNOM1:def 4;
      then b3 <= htp,T by TERMORD:def 6;
      then [b3,htp] in T by TERMORD:def 2;
      then [b9,b+htp] in T by A10,BAGORDER:def 5;
      hence b9 <= b+htp,T by TERMORD:def 2;
    end;
    (b*'p).(b+htp) = p.htp & b+htp is Element of Bags n by Lm9,PRE_POLY:def 12;
    then b + htp in Support(b*'p) by A7,POLYNOM1:def 4;
    hence thesis by A8,TERMORD:def 6;
  end;
end;
