9319 = (2 * 4659) + 1 ;
hence not 2 divides 9319 by NAT_4:9; :: thesis:
9319 = (3 * 3106) + 1 ;
hence not 3 divides 9319 by NAT_4:9; :: thesis:
9319 = (5 * 1863) + 4 ;
hence not 5 divides 9319 by NAT_4:9; :: thesis:
9319 = (7 * 1331) + 2 ;
hence not 7 divides 9319 by NAT_4:9; :: thesis:
9319 = (11 * 847) + 2 ;
hence not 11 divides 9319 by NAT_4:9; :: thesis:
9319 = (13 * 716) + 11 ;
hence not 13 divides 9319 by NAT_4:9; :: thesis:
9319 = (17 * 548) + 3 ;
hence not 17 divides 9319 by NAT_4:9; :: thesis:
9319 = (19 * 490) + 9 ;
hence not 19 divides 9319 by NAT_4:9; :: thesis:
9319 = (23 * 405) + 4 ;
hence not 23 divides 9319 by NAT_4:9; :: thesis:
9319 = (29 * 321) + 10 ;
hence not 29 divides 9319 by NAT_4:9; :: thesis:
9319 = (31 * 300) + 19 ;
hence not 31 divides 9319 by NAT_4:9; :: thesis:
9319 = (37 * 251) + 32 ;
hence not 37 divides 9319 by NAT_4:9; :: thesis:
9319 = (41 * 227) + 12 ;
hence not 41 divides 9319 by NAT_4:9; :: thesis:
9319 = (43 * 216) + 31 ;
hence not 43 divides 9319 by NAT_4:9; :: thesis:
9319 = (47 * 198) + 13 ;
hence not 47 divides 9319 by NAT_4:9; :: thesis:
9319 = (53 * 175) + 44 ;
hence not 53 divides 9319 by NAT_4:9; :: thesis:
9319 = (59 * 157) + 56 ;
hence not 59 divides 9319 by NAT_4:9; :: thesis:
9319 = (61 * 152) + 47 ;
hence not 61 divides 9319 by NAT_4:9; :: thesis:
9319 = (67 * 139) + 6 ;
hence not 67 divides 9319 by NAT_4:9; :: thesis:
9319 = (71 * 131) + 18 ;
hence not 71 divides 9319 by NAT_4:9; :: thesis:
9319 = (73 * 127) + 48 ;
hence not 73 divides 9319 by NAT_4:9; :: thesis:
9319 = (79 * 117) + 76 ;
hence not 79 divides 9319 by NAT_4:9; :: thesis:
9319 = (83 * 112) + 23 ;
hence not 83 divides 9319 by NAT_4:9; :: thesis:
9319 = (89 * 104) + 63 ;
hence not 89 divides 9319 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9319 & n is prime holds
not n divides 9319 by XPRIMET1:48;
hence 9319 is prime by NAT_4:14; :: thesis: