8971 = (2 * 4485) + 1 ;
hence not 2 divides 8971 by NAT_4:9; :: thesis:
8971 = (3 * 2990) + 1 ;
hence not 3 divides 8971 by NAT_4:9; :: thesis:
8971 = (5 * 1794) + 1 ;
hence not 5 divides 8971 by NAT_4:9; :: thesis:
8971 = (7 * 1281) + 4 ;
hence not 7 divides 8971 by NAT_4:9; :: thesis:
8971 = (11 * 815) + 6 ;
hence not 11 divides 8971 by NAT_4:9; :: thesis:
8971 = (13 * 690) + 1 ;
hence not 13 divides 8971 by NAT_4:9; :: thesis:
8971 = (17 * 527) + 12 ;
hence not 17 divides 8971 by NAT_4:9; :: thesis:
8971 = (19 * 472) + 3 ;
hence not 19 divides 8971 by NAT_4:9; :: thesis:
8971 = (23 * 390) + 1 ;
hence not 23 divides 8971 by NAT_4:9; :: thesis:
8971 = (29 * 309) + 10 ;
hence not 29 divides 8971 by NAT_4:9; :: thesis:
8971 = (31 * 289) + 12 ;
hence not 31 divides 8971 by NAT_4:9; :: thesis:
8971 = (37 * 242) + 17 ;
hence not 37 divides 8971 by NAT_4:9; :: thesis:
8971 = (41 * 218) + 33 ;
hence not 41 divides 8971 by NAT_4:9; :: thesis:
8971 = (43 * 208) + 27 ;
hence not 43 divides 8971 by NAT_4:9; :: thesis:
8971 = (47 * 190) + 41 ;
hence not 47 divides 8971 by NAT_4:9; :: thesis:
8971 = (53 * 169) + 14 ;
hence not 53 divides 8971 by NAT_4:9; :: thesis:
8971 = (59 * 152) + 3 ;
hence not 59 divides 8971 by NAT_4:9; :: thesis:
8971 = (61 * 147) + 4 ;
hence not 61 divides 8971 by NAT_4:9; :: thesis:
8971 = (67 * 133) + 60 ;
hence not 67 divides 8971 by NAT_4:9; :: thesis:
8971 = (71 * 126) + 25 ;
hence not 71 divides 8971 by NAT_4:9; :: thesis:
8971 = (73 * 122) + 65 ;
hence not 73 divides 8971 by NAT_4:9; :: thesis:
8971 = (79 * 113) + 44 ;
hence not 79 divides 8971 by NAT_4:9; :: thesis:
8971 = (83 * 108) + 7 ;
hence not 83 divides 8971 by NAT_4:9; :: thesis:
8971 = (89 * 100) + 71 ;
hence not 89 divides 8971 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8971 & n is prime holds
not n divides 8971 by XPRIMET1:48;
hence 8971 is prime by NAT_4:14; :: thesis: