8719 = (2 * 4359) + 1 ;
hence not 2 divides 8719 by NAT_4:9; :: thesis:
8719 = (3 * 2906) + 1 ;
hence not 3 divides 8719 by NAT_4:9; :: thesis:
8719 = (5 * 1743) + 4 ;
hence not 5 divides 8719 by NAT_4:9; :: thesis:
8719 = (7 * 1245) + 4 ;
hence not 7 divides 8719 by NAT_4:9; :: thesis:
8719 = (11 * 792) + 7 ;
hence not 11 divides 8719 by NAT_4:9; :: thesis:
8719 = (13 * 670) + 9 ;
hence not 13 divides 8719 by NAT_4:9; :: thesis:
8719 = (17 * 512) + 15 ;
hence not 17 divides 8719 by NAT_4:9; :: thesis:
8719 = (19 * 458) + 17 ;
hence not 19 divides 8719 by NAT_4:9; :: thesis:
8719 = (23 * 379) + 2 ;
hence not 23 divides 8719 by NAT_4:9; :: thesis:
8719 = (29 * 300) + 19 ;
hence not 29 divides 8719 by NAT_4:9; :: thesis:
8719 = (31 * 281) + 8 ;
hence not 31 divides 8719 by NAT_4:9; :: thesis:
8719 = (37 * 235) + 24 ;
hence not 37 divides 8719 by NAT_4:9; :: thesis:
8719 = (41 * 212) + 27 ;
hence not 41 divides 8719 by NAT_4:9; :: thesis:
8719 = (43 * 202) + 33 ;
hence not 43 divides 8719 by NAT_4:9; :: thesis:
8719 = (47 * 185) + 24 ;
hence not 47 divides 8719 by NAT_4:9; :: thesis:
8719 = (53 * 164) + 27 ;
hence not 53 divides 8719 by NAT_4:9; :: thesis:
8719 = (59 * 147) + 46 ;
hence not 59 divides 8719 by NAT_4:9; :: thesis:
8719 = (61 * 142) + 57 ;
hence not 61 divides 8719 by NAT_4:9; :: thesis:
8719 = (67 * 130) + 9 ;
hence not 67 divides 8719 by NAT_4:9; :: thesis:
8719 = (71 * 122) + 57 ;
hence not 71 divides 8719 by NAT_4:9; :: thesis:
8719 = (73 * 119) + 32 ;
hence not 73 divides 8719 by NAT_4:9; :: thesis:
8719 = (79 * 110) + 29 ;
hence not 79 divides 8719 by NAT_4:9; :: thesis:
8719 = (83 * 105) + 4 ;
hence not 83 divides 8719 by NAT_4:9; :: thesis:
8719 = (89 * 97) + 86 ;
hence not 89 divides 8719 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8719 & n is prime holds
not n divides 8719 by XPRIMET1:48;
hence 8719 is prime by NAT_4:14; :: thesis: