
theorem Th44:
  for L being non degenerated comRing, r being Element of L, p
being non-zero Polynomial of L st r is_a_root_of p holds len poly_quotient(p,r)
  > 0
proof
  let L be non degenerated comRing, r be Element of L, p be non-zero
  Polynomial of L such that
A1: r is_a_root_of p;
  assume len poly_quotient(p,r) <= 0;
  then
A2: len poly_quotient(p,r) = 0;
  len p > 0 by Th14;
  then len poly_quotient(p,r) + 1 = len p by A1,Def6;
  then Roots p = {} by A2,Th43;
  hence contradiction by A1,POLYNOM5:def 10;
end;
