8311 = (2 * 4155) + 1 ;
hence not 2 divides 8311 by NAT_4:9; :: thesis:
8311 = (3 * 2770) + 1 ;
hence not 3 divides 8311 by NAT_4:9; :: thesis:
8311 = (5 * 1662) + 1 ;
hence not 5 divides 8311 by NAT_4:9; :: thesis:
8311 = (7 * 1187) + 2 ;
hence not 7 divides 8311 by NAT_4:9; :: thesis:
8311 = (11 * 755) + 6 ;
hence not 11 divides 8311 by NAT_4:9; :: thesis:
8311 = (13 * 639) + 4 ;
hence not 13 divides 8311 by NAT_4:9; :: thesis:
8311 = (17 * 488) + 15 ;
hence not 17 divides 8311 by NAT_4:9; :: thesis:
8311 = (19 * 437) + 8 ;
hence not 19 divides 8311 by NAT_4:9; :: thesis:
8311 = (23 * 361) + 8 ;
hence not 23 divides 8311 by NAT_4:9; :: thesis:
8311 = (29 * 286) + 17 ;
hence not 29 divides 8311 by NAT_4:9; :: thesis:
8311 = (31 * 268) + 3 ;
hence not 31 divides 8311 by NAT_4:9; :: thesis:
8311 = (37 * 224) + 23 ;
hence not 37 divides 8311 by NAT_4:9; :: thesis:
8311 = (41 * 202) + 29 ;
hence not 41 divides 8311 by NAT_4:9; :: thesis:
8311 = (43 * 193) + 12 ;
hence not 43 divides 8311 by NAT_4:9; :: thesis:
8311 = (47 * 176) + 39 ;
hence not 47 divides 8311 by NAT_4:9; :: thesis:
8311 = (53 * 156) + 43 ;
hence not 53 divides 8311 by NAT_4:9; :: thesis:
8311 = (59 * 140) + 51 ;
hence not 59 divides 8311 by NAT_4:9; :: thesis:
8311 = (61 * 136) + 15 ;
hence not 61 divides 8311 by NAT_4:9; :: thesis:
8311 = (67 * 124) + 3 ;
hence not 67 divides 8311 by NAT_4:9; :: thesis:
8311 = (71 * 117) + 4 ;
hence not 71 divides 8311 by NAT_4:9; :: thesis:
8311 = (73 * 113) + 62 ;
hence not 73 divides 8311 by NAT_4:9; :: thesis:
8311 = (79 * 105) + 16 ;
hence not 79 divides 8311 by NAT_4:9; :: thesis:
8311 = (83 * 100) + 11 ;
hence not 83 divides 8311 by NAT_4:9; :: thesis:
8311 = (89 * 93) + 34 ;
hence not 89 divides 8311 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8311 & n is prime holds
not n divides 8311 by XPRIMET1:48;
hence 8311 is prime by NAT_4:14; :: thesis: