8329 = (2 * 4164) + 1 ;
hence not 2 divides 8329 by NAT_4:9; :: thesis:
8329 = (3 * 2776) + 1 ;
hence not 3 divides 8329 by NAT_4:9; :: thesis:
8329 = (5 * 1665) + 4 ;
hence not 5 divides 8329 by NAT_4:9; :: thesis:
8329 = (7 * 1189) + 6 ;
hence not 7 divides 8329 by NAT_4:9; :: thesis:
8329 = (11 * 757) + 2 ;
hence not 11 divides 8329 by NAT_4:9; :: thesis:
8329 = (13 * 640) + 9 ;
hence not 13 divides 8329 by NAT_4:9; :: thesis:
8329 = (17 * 489) + 16 ;
hence not 17 divides 8329 by NAT_4:9; :: thesis:
8329 = (19 * 438) + 7 ;
hence not 19 divides 8329 by NAT_4:9; :: thesis:
8329 = (23 * 362) + 3 ;
hence not 23 divides 8329 by NAT_4:9; :: thesis:
8329 = (29 * 287) + 6 ;
hence not 29 divides 8329 by NAT_4:9; :: thesis:
8329 = (31 * 268) + 21 ;
hence not 31 divides 8329 by NAT_4:9; :: thesis:
8329 = (37 * 225) + 4 ;
hence not 37 divides 8329 by NAT_4:9; :: thesis:
8329 = (41 * 203) + 6 ;
hence not 41 divides 8329 by NAT_4:9; :: thesis:
8329 = (43 * 193) + 30 ;
hence not 43 divides 8329 by NAT_4:9; :: thesis:
8329 = (47 * 177) + 10 ;
hence not 47 divides 8329 by NAT_4:9; :: thesis:
8329 = (53 * 157) + 8 ;
hence not 53 divides 8329 by NAT_4:9; :: thesis:
8329 = (59 * 141) + 10 ;
hence not 59 divides 8329 by NAT_4:9; :: thesis:
8329 = (61 * 136) + 33 ;
hence not 61 divides 8329 by NAT_4:9; :: thesis:
8329 = (67 * 124) + 21 ;
hence not 67 divides 8329 by NAT_4:9; :: thesis:
8329 = (71 * 117) + 22 ;
hence not 71 divides 8329 by NAT_4:9; :: thesis:
8329 = (73 * 114) + 7 ;
hence not 73 divides 8329 by NAT_4:9; :: thesis:
8329 = (79 * 105) + 34 ;
hence not 79 divides 8329 by NAT_4:9; :: thesis:
8329 = (83 * 100) + 29 ;
hence not 83 divides 8329 by NAT_4:9; :: thesis:
8329 = (89 * 93) + 52 ;
hence not 89 divides 8329 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8329 & n is prime holds
not n divides 8329 by XPRIMET1:48;
hence 8329 is prime by NAT_4:14; :: thesis: