
theorem Th41:
  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+1) c= Support Low(p,T,i)
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: i + 1 <= card(Support p) by NAT_1:13;
  then
A3: card Support p - i >= 1 by XREAL_1:19;
A4: Support Low(p,T,i) = Lower_Support(p,T,i) by A1,Lm3;
  then card Support l = card(Support p) - i by A1,Th24;
  then
A5: HT(Low(p,T,i),T) in Lower_Support(p,T,i) by A3,A4,CARD_1:27,TERMORD:def 6;
A6: HT(Low(p,T,i+1),T) <= HT(Low(p,T,i),T), T by A1,Th38;
A7: Support(Low(p,T,i+1)) c= Support(p) by A2,Th26;
    let u9 be object;
    assume
A8: u9 in Support l1;
    then reconsider u = u9 as Element of Bags n;
    u <= HT(Low(p,T,i+1),T),T by A8,TERMORD:def 6;
    hence u9 in Support l by A1,A7,A4,A6,A5,A8,Th24,TERMORD:8;
end;
