8231 = (2 * 4115) + 1 ;
hence not 2 divides 8231 by NAT_4:9; :: thesis:
8231 = (3 * 2743) + 2 ;
hence not 3 divides 8231 by NAT_4:9; :: thesis:
8231 = (5 * 1646) + 1 ;
hence not 5 divides 8231 by NAT_4:9; :: thesis:
8231 = (7 * 1175) + 6 ;
hence not 7 divides 8231 by NAT_4:9; :: thesis:
8231 = (11 * 748) + 3 ;
hence not 11 divides 8231 by NAT_4:9; :: thesis:
8231 = (13 * 633) + 2 ;
hence not 13 divides 8231 by NAT_4:9; :: thesis:
8231 = (17 * 484) + 3 ;
hence not 17 divides 8231 by NAT_4:9; :: thesis:
8231 = (19 * 433) + 4 ;
hence not 19 divides 8231 by NAT_4:9; :: thesis:
8231 = (23 * 357) + 20 ;
hence not 23 divides 8231 by NAT_4:9; :: thesis:
8231 = (29 * 283) + 24 ;
hence not 29 divides 8231 by NAT_4:9; :: thesis:
8231 = (31 * 265) + 16 ;
hence not 31 divides 8231 by NAT_4:9; :: thesis:
8231 = (37 * 222) + 17 ;
hence not 37 divides 8231 by NAT_4:9; :: thesis:
8231 = (41 * 200) + 31 ;
hence not 41 divides 8231 by NAT_4:9; :: thesis:
8231 = (43 * 191) + 18 ;
hence not 43 divides 8231 by NAT_4:9; :: thesis:
8231 = (47 * 175) + 6 ;
hence not 47 divides 8231 by NAT_4:9; :: thesis:
8231 = (53 * 155) + 16 ;
hence not 53 divides 8231 by NAT_4:9; :: thesis:
8231 = (59 * 139) + 30 ;
hence not 59 divides 8231 by NAT_4:9; :: thesis:
8231 = (61 * 134) + 57 ;
hence not 61 divides 8231 by NAT_4:9; :: thesis:
8231 = (67 * 122) + 57 ;
hence not 67 divides 8231 by NAT_4:9; :: thesis:
8231 = (71 * 115) + 66 ;
hence not 71 divides 8231 by NAT_4:9; :: thesis:
8231 = (73 * 112) + 55 ;
hence not 73 divides 8231 by NAT_4:9; :: thesis:
8231 = (79 * 104) + 15 ;
hence not 79 divides 8231 by NAT_4:9; :: thesis:
8231 = (83 * 99) + 14 ;
hence not 83 divides 8231 by NAT_4:9; :: thesis:
8231 = (89 * 92) + 43 ;
hence not 89 divides 8231 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8231 & n is prime holds
not n divides 8231 by XPRIMET1:48;
hence 8231 is prime by NAT_4:14; :: thesis: