
theorem
  1409 is prime
proof
  now
    1409 = 2*704 + 1; hence not 2 divides 1409 by NAT_4:9;
    1409 = 3*469 + 2; hence not 3 divides 1409 by NAT_4:9;
    1409 = 5*281 + 4; hence not 5 divides 1409 by NAT_4:9;
    1409 = 7*201 + 2; hence not 7 divides 1409 by NAT_4:9;
    1409 = 11*128 + 1; hence not 11 divides 1409 by NAT_4:9;
    1409 = 13*108 + 5; hence not 13 divides 1409 by NAT_4:9;
    1409 = 17*82 + 15; hence not 17 divides 1409 by NAT_4:9;
    1409 = 19*74 + 3; hence not 19 divides 1409 by NAT_4:9;
    1409 = 23*61 + 6; hence not 23 divides 1409 by NAT_4:9;
    1409 = 29*48 + 17; hence not 29 divides 1409 by NAT_4:9;
    1409 = 31*45 + 14; hence not 31 divides 1409 by NAT_4:9;
    1409 = 37*38 + 3; hence not 37 divides 1409 by NAT_4:9;
  end;
  then for n being Element of NAT st 1 < n & n*n <= 1409 & n is prime
  holds not n divides 1409 by XPRIMET1:24;
  hence thesis by NAT_4:14;
end;
