
theorem Th23:
  for n, m, k being Element of NAT st PFArt (n,m) c= PFArt (k,m) holds n = k
proof
  let n, m, k be Element of NAT;
  assume
A1: PFArt (n,m) c= PFArt (k,m);
  assume n <> k;
  then 2*n <> 2*k;
  then
A2: [2*n,m] <> [2*k,m] by XTUPLE_0:1;
  [2*n,m] in PFArt (n,m) by Def2;
  then ex m9 being odd Element of NAT st m9 <= 2*k & [m9,m] = [ 2*n,m] by A1,A2
,Def2;
  hence thesis by XTUPLE_0:1;
end;
