8297 = (2 * 4148) + 1 ;
hence not 2 divides 8297 by NAT_4:9; :: thesis:
8297 = (3 * 2765) + 2 ;
hence not 3 divides 8297 by NAT_4:9; :: thesis:
8297 = (5 * 1659) + 2 ;
hence not 5 divides 8297 by NAT_4:9; :: thesis:
8297 = (7 * 1185) + 2 ;
hence not 7 divides 8297 by NAT_4:9; :: thesis:
8297 = (11 * 754) + 3 ;
hence not 11 divides 8297 by NAT_4:9; :: thesis:
8297 = (13 * 638) + 3 ;
hence not 13 divides 8297 by NAT_4:9; :: thesis:
8297 = (17 * 488) + 1 ;
hence not 17 divides 8297 by NAT_4:9; :: thesis:
8297 = (19 * 436) + 13 ;
hence not 19 divides 8297 by NAT_4:9; :: thesis:
8297 = (23 * 360) + 17 ;
hence not 23 divides 8297 by NAT_4:9; :: thesis:
8297 = (29 * 286) + 3 ;
hence not 29 divides 8297 by NAT_4:9; :: thesis:
8297 = (31 * 267) + 20 ;
hence not 31 divides 8297 by NAT_4:9; :: thesis:
8297 = (37 * 224) + 9 ;
hence not 37 divides 8297 by NAT_4:9; :: thesis:
8297 = (41 * 202) + 15 ;
hence not 41 divides 8297 by NAT_4:9; :: thesis:
8297 = (43 * 192) + 41 ;
hence not 43 divides 8297 by NAT_4:9; :: thesis:
8297 = (47 * 176) + 25 ;
hence not 47 divides 8297 by NAT_4:9; :: thesis:
8297 = (53 * 156) + 29 ;
hence not 53 divides 8297 by NAT_4:9; :: thesis:
8297 = (59 * 140) + 37 ;
hence not 59 divides 8297 by NAT_4:9; :: thesis:
8297 = (61 * 136) + 1 ;
hence not 61 divides 8297 by NAT_4:9; :: thesis:
8297 = (67 * 123) + 56 ;
hence not 67 divides 8297 by NAT_4:9; :: thesis:
8297 = (71 * 116) + 61 ;
hence not 71 divides 8297 by NAT_4:9; :: thesis:
8297 = (73 * 113) + 48 ;
hence not 73 divides 8297 by NAT_4:9; :: thesis:
8297 = (79 * 105) + 2 ;
hence not 79 divides 8297 by NAT_4:9; :: thesis:
8297 = (83 * 99) + 80 ;
hence not 83 divides 8297 by NAT_4:9; :: thesis:
8297 = (89 * 93) + 20 ;
hence not 89 divides 8297 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8297 & n is prime holds
not n divides 8297 by XPRIMET1:48;
hence 8297 is prime by NAT_4:14; :: thesis: