
theorem Th42:
  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 Support Low(p,T,i) \ Support Low(p,T,i+1) = {HT(Low(p,T,i),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 l = Low(p,T,i), l1 = Low(p,T,i+1);
  assume
A1: i < card(Support p);
  then
A2: Support Low(p,T,i) = Lower_Support(p,T,i) by Lm3;
  then
A3: card Support l = card(Support p) - i by A1,Th24;
  now
    assume Lower_Support(p,T,i) = {};
    then card(Support p) - i = 0 by A1,Th24,CARD_1:27;
    hence contradiction by A1;
  end;
  then
A4: HT(Low(p,T,i),T) in Support l by A2,TERMORD:def 6;
A5: Support(Low(p,T,i)) c= Support(p) by A1,Th26;
A6: i + 1 <= card(Support p) by A1,NAT_1:13;
  then Support Low(p,T,i+1) = Lower_Support(p,T,i+1) by Lm3;
  then
A7: card Support l1 = card(Support p) - (i+1) by A6,Th24;
  then
  card(Support Low(p,T,i) \ Support Low(p,T,i+1)) = (card(Support p) - i)
  - (card(Support p) - (i+1)) by A1,A3,Th41,CARD_2:44
    .= 1;
  then consider x being object such that
A8: Support Low(p,T,i) \ Support Low(p,T,i+1) = {x} by CARD_2:42;
A9: Support Low(p,T,i+1) = Lower_Support(p,T,i+1) by A6,Lm3;
  now
    assume
A10: x <> HT(Low(p,T,i),T);
A11: now
      assume not HT(Low(p,T,i),T) in Support l1;
      then HT(Low(p,T,i),T) in Support l \ Support l1 by A4,XBOOLE_0:def 5;
      hence contradiction by A8,A10,TARSKI:def 1;
    end;
A12: now
      let u be object;
      assume
A13:  u in Support l;
      then reconsider u9 = u as Element of Bags n;
      u9 <= HT(Low(p,T,i),T),T by A13,TERMORD:def 6;
      hence u in Support l1 by A6,A5,A9,A11,A13,Th24;
    end;
    Support Low(p,T,i+1) c= Support Low(p,T,i) by A1,Th41;
    then for u being object holds u in Support l1 implies u in Support l;
    then card(Support p) + -i <= card(Support p) + -(i+1) by A3,A7,A12,TARSKI:2
;
    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 A8;
end;
