8731 = (2 * 4365) + 1 ;
hence not 2 divides 8731 by NAT_4:9; :: thesis:
8731 = (3 * 2910) + 1 ;
hence not 3 divides 8731 by NAT_4:9; :: thesis:
8731 = (5 * 1746) + 1 ;
hence not 5 divides 8731 by NAT_4:9; :: thesis:
8731 = (7 * 1247) + 2 ;
hence not 7 divides 8731 by NAT_4:9; :: thesis:
8731 = (11 * 793) + 8 ;
hence not 11 divides 8731 by NAT_4:9; :: thesis:
8731 = (13 * 671) + 8 ;
hence not 13 divides 8731 by NAT_4:9; :: thesis:
8731 = (17 * 513) + 10 ;
hence not 17 divides 8731 by NAT_4:9; :: thesis:
8731 = (19 * 459) + 10 ;
hence not 19 divides 8731 by NAT_4:9; :: thesis:
8731 = (23 * 379) + 14 ;
hence not 23 divides 8731 by NAT_4:9; :: thesis:
8731 = (29 * 301) + 2 ;
hence not 29 divides 8731 by NAT_4:9; :: thesis:
8731 = (31 * 281) + 20 ;
hence not 31 divides 8731 by NAT_4:9; :: thesis:
8731 = (37 * 235) + 36 ;
hence not 37 divides 8731 by NAT_4:9; :: thesis:
8731 = (41 * 212) + 39 ;
hence not 41 divides 8731 by NAT_4:9; :: thesis:
8731 = (43 * 203) + 2 ;
hence not 43 divides 8731 by NAT_4:9; :: thesis:
8731 = (47 * 185) + 36 ;
hence not 47 divides 8731 by NAT_4:9; :: thesis:
8731 = (53 * 164) + 39 ;
hence not 53 divides 8731 by NAT_4:9; :: thesis:
8731 = (59 * 147) + 58 ;
hence not 59 divides 8731 by NAT_4:9; :: thesis:
8731 = (61 * 143) + 8 ;
hence not 61 divides 8731 by NAT_4:9; :: thesis:
8731 = (67 * 130) + 21 ;
hence not 67 divides 8731 by NAT_4:9; :: thesis:
8731 = (71 * 122) + 69 ;
hence not 71 divides 8731 by NAT_4:9; :: thesis:
8731 = (73 * 119) + 44 ;
hence not 73 divides 8731 by NAT_4:9; :: thesis:
8731 = (79 * 110) + 41 ;
hence not 79 divides 8731 by NAT_4:9; :: thesis:
8731 = (83 * 105) + 16 ;
hence not 83 divides 8731 by NAT_4:9; :: thesis:
8731 = (89 * 98) + 9 ;
hence not 89 divides 8731 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8731 & n is prime holds
not n divides 8731 by XPRIMET1:48;
hence 8731 is prime by NAT_4:14; :: thesis: