
theorem Th38:
  for n being Ordinal, T being connected admissible 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 i < card(
  Support p) holds HT(Low(p,T,i+1),T) <= HT(Low(p,T,i),T), T
proof
  let n be Ordinal, T be connected admissible 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;
  set li = Low(p,T,i), li1 = Low(p,T,i+1);
  assume
A1: i < card(Support p);
  then Support li = Lower_Support(p,T,i) by Lm3;
  then
A2: card Support li = card(Support p) - i by A1,Th24;
A3: i + 1 <= card(Support p) by A1,NAT_1:13;
  then
A4: Support li1 = Lower_Support(p,T,i+1) by Lm3;
  then
A5: card Support li1 = card(Support p) - (i+1) by A3,Th24;
A6: Support li c= Support(p) by A1,Th26;
  now
    per cases;
    case
      i = card(Support p) - 1;
      then card Support li1 = card(Support p) - card(Support p) by A4,Th24
        .= 0;
      then Support li1 = {};
      then HT(li1,T) = EmptyBag n by TERMORD:def 6;
      hence thesis by TERMORD:9;
    end;
    case
      i <> card(Support p) - 1;
      then card Lower_Support(p,T,i+1) <> 0 by A4,A5;
      then Lower_Support(p,T,i+1) <> {};
      then
A7:   HT(Low(p,T,i+1),T) in Lower_Support(p,T,i+1) by A4,TERMORD:def 6;
      now
        assume HT(Low(p,T,i),T) < HT(Low(p,T,i+1),T), T;
        then
A8:     HT(Low(p,T,i),T) <= HT(Low(p,T,i+1),T), T by TERMORD:def 3;
        now
          let u9 be object;
          assume
A9:       u9 in Support li;
          then reconsider u = u9 as Element of Bags n;
          u <= HT(Low(p,T,i),T),T by A9,TERMORD:def 6;
          hence u9 in Support li1 by A3,A6,A4,A7,A8,A9,Th24,TERMORD:8;
        end;
        then Support li c= Support li1;
        then card(Support p) + -i <= card(Support p) + -(i+1) by A2,A5,NAT_1:43
;
        then - i <= -(i + 1) by XREAL_1:6;
        then i + 1 <= i by XREAL_1:24;
        then (i + 1) - i <= i - i by XREAL_1:9;
        then 1 <= 0;
        hence contradiction;
      end;
      hence thesis by TERMORD:5;
    end;
  end;
  hence thesis;
end;
