
theorem
  419 is prime
proof
  now
    419 = 2*209 + 1; hence not 2 divides 419 by NAT_4:9;
    419 = 3*139 + 2; hence not 3 divides 419 by NAT_4:9;
    419 = 5*83 + 4; hence not 5 divides 419 by NAT_4:9;
    419 = 7*59 + 6; hence not 7 divides 419 by NAT_4:9;
    419 = 11*38 + 1; hence not 11 divides 419 by NAT_4:9;
    419 = 13*32 + 3; hence not 13 divides 419 by NAT_4:9;
    419 = 17*24 + 11; hence not 17 divides 419 by NAT_4:9;
    419 = 19*22 + 1; hence not 19 divides 419 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 419 & n is prime
  holds not n divides 419 by XPRIMET1:16;
  hence thesis by NAT_4:14;
end;
