
theorem
  for n, m, k being Element of NAT st n <= k holds PFCrt (n,m) c= PFCrt (k,m)
proof
  let n, m, k be Element of NAT;
  assume n <= k;
  then 2*n <= 2*k by NAT_1:4;
  then
A1: 2*n + 1 <= 2*k + 1 by XREAL_1:6;
  let x be object;
  assume x in PFCrt (n, m);
  then consider m9 being odd Element of NAT such that
A2: m9 <= 2*n + 1 and
A3: [m9,m] = x by Def3;
  m9 <= 2*k + 1 by A1,A2,XXREAL_0:2;
  hence thesis by A3,Def3;
end;
