
theorem Th37:
  for n being Ordinal, T being connected TermOrder of n, L being
  add-associative right_zeroed right_complementable non trivial addLoopStr, p
  being Polynomial of n,L, j being Element of NAT st j = card(Support p) - 1
  holds Low(p,T,j) is non-zero Monomial of n,L
proof
  let n be Ordinal, T be connected TermOrder of n, L be add-associative
right_zeroed right_complementable non trivial addLoopStr, p be Polynomial of
  n,L, j be Element of NAT;
  set l = Low(p,T,j);
  assume
A1: j = card(Support p) - 1;
A2: now
    assume j > card(Support p);
    then card(Support p) - 1 + 1 > card(Support p) + 1 by A1,XREAL_1:8;
    then card(Support p) + -card(Support p) > (card(Support p) + 1) + -card(
    Support p) by XREAL_1:8;
    hence contradiction;
  end;
  then Support l = Lower_Support(p,T,j) by Lm3;
  then card(Support l) = card(Support p) - (card(Support p) - 1) by A1,A2,Th24;
  then consider x being object such that
A3: Support l = {x} by CARD_2:42;
  x in Support l by A3,TARSKI:def 1;
  then
A4: x is Element of Bags n;
  l <> 0_(n,L) by A3,POLYNOM7:1;
  hence thesis by A3,A4,POLYNOM7:6,def 1;
end;
