 reserve o for object;
 reserve F for non almost_trivial Field;
 reserve x,a for Element of F;
reserve n for non zero Nat;

theorem Th16:
  for R being flat Ring, p being Polynomial of R holds not p in [#]R
  proof
    let R be flat Ring, p be Polynomial of R;
    now assume
A1:   p in [#]R; then
      reconsider a = p as Element of R;
A2:   the_rank_of p = the_rank_of (p.0) by A1,Def10;
      dom p = NAT by FUNCT_2:def 1; then
A3:   [0,p.0] in p by FUNCT_1:def 2;
      the_rank_of (p.0) in the_rank_of [0,p.0] by CLASSES1:84;
      hence contradiction by A2,A3,CLASSES1:68;
    end;
    hence thesis;
  end;
