8089 = (2 * 4044) + 1 ;
hence not 2 divides 8089 by NAT_4:9; :: thesis:
8089 = (3 * 2696) + 1 ;
hence not 3 divides 8089 by NAT_4:9; :: thesis:
8089 = (5 * 1617) + 4 ;
hence not 5 divides 8089 by NAT_4:9; :: thesis:
8089 = (7 * 1155) + 4 ;
hence not 7 divides 8089 by NAT_4:9; :: thesis:
8089 = (11 * 735) + 4 ;
hence not 11 divides 8089 by NAT_4:9; :: thesis:
8089 = (13 * 622) + 3 ;
hence not 13 divides 8089 by NAT_4:9; :: thesis:
8089 = (17 * 475) + 14 ;
hence not 17 divides 8089 by NAT_4:9; :: thesis:
8089 = (19 * 425) + 14 ;
hence not 19 divides 8089 by NAT_4:9; :: thesis:
8089 = (23 * 351) + 16 ;
hence not 23 divides 8089 by NAT_4:9; :: thesis:
8089 = (29 * 278) + 27 ;
hence not 29 divides 8089 by NAT_4:9; :: thesis:
8089 = (31 * 260) + 29 ;
hence not 31 divides 8089 by NAT_4:9; :: thesis:
8089 = (37 * 218) + 23 ;
hence not 37 divides 8089 by NAT_4:9; :: thesis:
8089 = (41 * 197) + 12 ;
hence not 41 divides 8089 by NAT_4:9; :: thesis:
8089 = (43 * 188) + 5 ;
hence not 43 divides 8089 by NAT_4:9; :: thesis:
8089 = (47 * 172) + 5 ;
hence not 47 divides 8089 by NAT_4:9; :: thesis:
8089 = (53 * 152) + 33 ;
hence not 53 divides 8089 by NAT_4:9; :: thesis:
8089 = (59 * 137) + 6 ;
hence not 59 divides 8089 by NAT_4:9; :: thesis:
8089 = (61 * 132) + 37 ;
hence not 61 divides 8089 by NAT_4:9; :: thesis:
8089 = (67 * 120) + 49 ;
hence not 67 divides 8089 by NAT_4:9; :: thesis:
8089 = (71 * 113) + 66 ;
hence not 71 divides 8089 by NAT_4:9; :: thesis:
8089 = (73 * 110) + 59 ;
hence not 73 divides 8089 by NAT_4:9; :: thesis:
8089 = (79 * 102) + 31 ;
hence not 79 divides 8089 by NAT_4:9; :: thesis:
8089 = (83 * 97) + 38 ;
hence not 83 divides 8089 by NAT_4:9; :: thesis:
8089 = (89 * 90) + 79 ;
hence not 89 divides 8089 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8089 & n is prime holds
not n divides 8089 by XPRIMET1:48;
hence 8089 is prime by NAT_4:14; :: thesis: