now :: thesis: ( not 2 divides 317 & not 3 divides 317 & not 5 divides 317 & not 7 divides 317 & not 11 divides 317 & not 13 divides 317 & not 17 divides 317 )

317 = (2 * 158) + 1 ;
hence not 2 divides 317 by NAT_4:9; :: thesis: ( not 3 divides 317 & not 5 divides 317 & not 7 divides 317 & not 11 divides 317 & not 13 divides 317 & not 17 divides 317 )
317 = (3 * 105) + 2 ;
hence not 3 divides 317 by NAT_4:9; :: thesis: ( not 5 divides 317 & not 7 divides 317 & not 11 divides 317 & not 13 divides 317 & not 17 divides 317 )
317 = (5 * 63) + 2 ;
hence not 5 divides 317 by NAT_4:9; :: thesis: ( not 7 divides 317 & not 11 divides 317 & not 13 divides 317 & not 17 divides 317 )
317 = (7 * 45) + 2 ;
hence not 7 divides 317 by NAT_4:9; :: thesis: ( not 11 divides 317 & not 13 divides 317 & not 17 divides 317 )
317 = (11 * 28) + 9 ;
hence not 11 divides 317 by NAT_4:9; :: thesis: ( not 13 divides 317 & not 17 divides 317 )
317 = (13 * 24) + 5 ;
hence not 13 divides 317 by NAT_4:9; :: thesis: not 17 divides 317
317 = (17 * 18) + 11 ;
hence not 17 divides 317 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 317 & n is prime holds
not n divides 317 by XPRIMET1:14;
hence 317 is prime by NAT_4:14; :: thesis: verum