2297 = (2 * 1148) + 1 ;
hence not 2 divides 2297 by NAT_4:9; :: thesis:
2297 = (3 * 765) + 2 ;
hence not 3 divides 2297 by NAT_4:9; :: thesis:
2297 = (5 * 459) + 2 ;
hence not 5 divides 2297 by NAT_4:9; :: thesis:
2297 = (7 * 328) + 1 ;
hence not 7 divides 2297 by NAT_4:9; :: thesis:
2297 = (11 * 208) + 9 ;
hence not 11 divides 2297 by NAT_4:9; :: thesis:
2297 = (13 * 176) + 9 ;
hence not 13 divides 2297 by NAT_4:9; :: thesis:
2297 = (17 * 135) + 2 ;
hence not 17 divides 2297 by NAT_4:9; :: thesis:
2297 = (19 * 120) + 17 ;
hence not 19 divides 2297 by NAT_4:9; :: thesis:
2297 = (23 * 99) + 20 ;
hence not 23 divides 2297 by NAT_4:9; :: thesis:
2297 = (29 * 79) + 6 ;
hence not 29 divides 2297 by NAT_4:9; :: thesis:
2297 = (31 * 74) + 3 ;
hence not 31 divides 2297 by NAT_4:9; :: thesis:
2297 = (37 * 62) + 3 ;
hence not 37 divides 2297 by NAT_4:9; :: thesis:
2297 = (41 * 56) + 1 ;
hence not 41 divides 2297 by NAT_4:9; :: thesis:
2297 = (43 * 53) + 18 ;
hence not 43 divides 2297 by NAT_4:9; :: thesis:
2297 = (47 * 48) + 41 ;
hence not 47 divides 2297 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2297 & n is prime holds
not n divides 2297 by XPRIMET1:30;
hence 2297 is prime by NAT_4:14; :: thesis: