2309 = (2 * 1154) + 1 ;
hence not 2 divides 2309 by NAT_4:9; :: thesis:
2309 = (3 * 769) + 2 ;
hence not 3 divides 2309 by NAT_4:9; :: thesis:
2309 = (5 * 461) + 4 ;
hence not 5 divides 2309 by NAT_4:9; :: thesis:
2309 = (7 * 329) + 6 ;
hence not 7 divides 2309 by NAT_4:9; :: thesis:
2309 = (11 * 209) + 10 ;
hence not 11 divides 2309 by NAT_4:9; :: thesis:
2309 = (13 * 177) + 8 ;
hence not 13 divides 2309 by NAT_4:9; :: thesis:
2309 = (17 * 135) + 14 ;
hence not 17 divides 2309 by NAT_4:9; :: thesis:
2309 = (19 * 121) + 10 ;
hence not 19 divides 2309 by NAT_4:9; :: thesis:
2309 = (23 * 100) + 9 ;
hence not 23 divides 2309 by NAT_4:9; :: thesis:
2309 = (29 * 79) + 18 ;
hence not 29 divides 2309 by NAT_4:9; :: thesis:
2309 = (31 * 74) + 15 ;
hence not 31 divides 2309 by NAT_4:9; :: thesis:
2309 = (37 * 62) + 15 ;
hence not 37 divides 2309 by NAT_4:9; :: thesis:
2309 = (41 * 56) + 13 ;
hence not 41 divides 2309 by NAT_4:9; :: thesis:
2309 = (43 * 53) + 30 ;
hence not 43 divides 2309 by NAT_4:9; :: thesis:
2309 = (47 * 49) + 6 ;
hence not 47 divides 2309 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2309 & n is prime holds
not n divides 2309 by XPRIMET1:30;
hence 2309 is prime by NAT_4:14; :: thesis: