
theorem Th17:
  for n, k being Element of NAT holds PFCrt (n,k) c< PFCrt (n+1,k)
proof
  let n,k be Element of NAT;
  thus PFCrt (n,k) c= PFCrt (n+1,k)
  proof
    let x be object;
    assume x in PFCrt (n,k);
    then consider m being odd Element of NAT such that
A1: m <= 2*n + 1 and
A2: [m,k] = x by Def3;
    2*n + 1 <= 2*(n+1) + 1 by Lm4;
    then m <= 2*(n+1) + 1 by A1,XXREAL_0:2;
    hence thesis by A2,Def3;
  end;
  thus thesis by Lm5;
end;
