
theorem Th18:
  for n, m, k being Element of NAT holds not PFArt (n, m) c= PFCrt (k, m)
proof
  let n, m, k be Element of NAT;
  set x = [2*n,m];
A1: not x in PFCrt (k,m)
  proof
    assume x in PFCrt (k,m);
    then ex m9 being odd Element of NAT st m9 <= 2*k + 1 & [m9,m] = x by Def3;
    hence thesis by XTUPLE_0:1;
  end;
  x in PFArt (n, m) by Def2;
  hence thesis by A1;
end;
