
theorem Th15:
  for n, k being Element of NAT holds PFCrt (n,k) misses {[2*n+3,k ]}
proof
  let n, k be Element of NAT;
  assume PFCrt (n,k) /\ {[2*n+3,k]} <> {};
  then consider x being object such that
A1: x in PFCrt (n,k) /\ {[2*n+3,k]} by XBOOLE_0:def 1;
  x in PFCrt (n,k) by A1,XBOOLE_0:def 4;
  then consider m being odd Element of NAT such that
A2: m <= 2*n + 1 and
A3: [m,k] = x by Def3;
  x in {[2*n+3,k]} by A1,XBOOLE_0:def 4;
  then x = [2*n+3,k] by TARSKI:def 1;
  then m = 2*n+3 by A3,XTUPLE_0:1;
  hence thesis by A2,XREAL_1:6;
end;
