8597 = (2 * 4298) + 1 ;
hence not 2 divides 8597 by NAT_4:9; :: thesis:
8597 = (3 * 2865) + 2 ;
hence not 3 divides 8597 by NAT_4:9; :: thesis:
8597 = (5 * 1719) + 2 ;
hence not 5 divides 8597 by NAT_4:9; :: thesis:
8597 = (7 * 1228) + 1 ;
hence not 7 divides 8597 by NAT_4:9; :: thesis:
8597 = (11 * 781) + 6 ;
hence not 11 divides 8597 by NAT_4:9; :: thesis:
8597 = (13 * 661) + 4 ;
hence not 13 divides 8597 by NAT_4:9; :: thesis:
8597 = (17 * 505) + 12 ;
hence not 17 divides 8597 by NAT_4:9; :: thesis:
8597 = (19 * 452) + 9 ;
hence not 19 divides 8597 by NAT_4:9; :: thesis:
8597 = (23 * 373) + 18 ;
hence not 23 divides 8597 by NAT_4:9; :: thesis:
8597 = (29 * 296) + 13 ;
hence not 29 divides 8597 by NAT_4:9; :: thesis:
8597 = (31 * 277) + 10 ;
hence not 31 divides 8597 by NAT_4:9; :: thesis:
8597 = (37 * 232) + 13 ;
hence not 37 divides 8597 by NAT_4:9; :: thesis:
8597 = (41 * 209) + 28 ;
hence not 41 divides 8597 by NAT_4:9; :: thesis:
8597 = (43 * 199) + 40 ;
hence not 43 divides 8597 by NAT_4:9; :: thesis:
8597 = (47 * 182) + 43 ;
hence not 47 divides 8597 by NAT_4:9; :: thesis:
8597 = (53 * 162) + 11 ;
hence not 53 divides 8597 by NAT_4:9; :: thesis:
8597 = (59 * 145) + 42 ;
hence not 59 divides 8597 by NAT_4:9; :: thesis:
8597 = (61 * 140) + 57 ;
hence not 61 divides 8597 by NAT_4:9; :: thesis:
8597 = (67 * 128) + 21 ;
hence not 67 divides 8597 by NAT_4:9; :: thesis:
8597 = (71 * 121) + 6 ;
hence not 71 divides 8597 by NAT_4:9; :: thesis:
8597 = (73 * 117) + 56 ;
hence not 73 divides 8597 by NAT_4:9; :: thesis:
8597 = (79 * 108) + 65 ;
hence not 79 divides 8597 by NAT_4:9; :: thesis:
8597 = (83 * 103) + 48 ;
hence not 83 divides 8597 by NAT_4:9; :: thesis:
8597 = (89 * 96) + 53 ;
hence not 89 divides 8597 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8597 & n is prime holds
not n divides 8597 by XPRIMET1:48;
hence 8597 is prime by NAT_4:14; :: thesis: