now :: thesis: for n being Element of NAT st 1 < n & n * n <= 317 & n is prime holds
not n divides 317

let n be Element of NAT ; :: thesis: ( 1 < n & n * n <= 317 & n is prime implies not n divides 317 )
317 = (2 * 158) + 1 ;
then A1: not 2 divides 317 by Th9;
317 = (3 * 105) + 2 ;
then A2: not 3 divides 317 by Th9;
317 = (13 * 24) + 5 ;
then A3: not 13 divides 317 by Th9;
317 = (11 * 28) + 9 ;
then A4: not 11 divides 317 by Th9;
317 = (19 * 16) + 13 ;
then A5: not 19 divides 317 by Th9;
317 = (17 * 18) + 11 ;
then A6: not 17 divides 317 by Th9;
317 = (23 * 13) + 18 ;
then A7: not 23 divides 317 by Th9;
317 = (7 * 45) + 2 ;
then A8: not 7 divides 317 by Th9;
317 = (5 * 63) + 2 ;
then A9: not 5 divides 317 by Th9;
assume ( 1 < n & n * n <= 317 & n is prime ) ; :: thesis: not n divides 317
hence not n divides 317 by A1, A2, A9, A8, A4, A3, A6, A5, A7, Lm6; :: thesis: verum

end;

hence 317 is prime by Th14; :: thesis: verum