
theorem Th39:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non empty addLoopStr, p
being Polynomial of n,L, i being Element of NAT st 0 < i & i < card(Support p)
  holds HT(Low(p,T,i),T) < HT(p,T),T
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_zeroed right_complementable non empty addLoopStr, p be Polynomial of n,
  L, i be Element of NAT;
  assume that
A1: 0 < i and
A2: i < card(Support p);
  set l = Low(p,T,i);
  now
    per cases;
    case
      l = 0_(n,L);
      then
A3:   card Support l = 0 by CARD_1:27,POLYNOM7:1;
      Support l = Lower_Support(p,T,i) by A2,Lm3;
      then 0 + i = card(Support p) - i + i by A2,A3,Th24;
      hence contradiction by A2;
    end;
    case
A4:   l <> 0_(n,L);
A5:   Support(Low(p,T,i)) c= Support(p) by A2,Th26;
A6:   Support Low(p,T,i) = Lower_Support(p,T,i) by A2,Lm3;
A7:   now
        assume
A8:     HT(p,T) in Support l;
A9:     now
          let u be object;
          assume
A10:      u in Support p;
          then reconsider x = u as Element of Bags n;
          x <= HT(p,T),T by A10,TERMORD:def 6;
          hence u in Support l by A2,A6,A8,A10,Th24;
        end;
        for u being object holds u in Support l implies u in Support p by A5;
        then card Support p = card Support l by A9,TARSKI:2
          .= card(Support p) - i by A2,A6,Th24;
        hence contradiction by A1;
      end;
      Support l <> {} by A4,POLYNOM7:1;
      then
A11:  HT(l,T) in Support l by TERMORD:def 6;
      then HT(l,T) <= HT(p,T),T by A5,TERMORD:def 6;
      hence thesis by A7,A11,TERMORD:def 3;
    end;
  end;
  hence thesis;
end;
