2333 = (2 * 1166) + 1 ;
hence not 2 divides 2333 by NAT_4:9; :: thesis:
2333 = (3 * 777) + 2 ;
hence not 3 divides 2333 by NAT_4:9; :: thesis:
2333 = (5 * 466) + 3 ;
hence not 5 divides 2333 by NAT_4:9; :: thesis:
2333 = (7 * 333) + 2 ;
hence not 7 divides 2333 by NAT_4:9; :: thesis:
2333 = (11 * 212) + 1 ;
hence not 11 divides 2333 by NAT_4:9; :: thesis:
2333 = (13 * 179) + 6 ;
hence not 13 divides 2333 by NAT_4:9; :: thesis:
2333 = (17 * 137) + 4 ;
hence not 17 divides 2333 by NAT_4:9; :: thesis:
2333 = (19 * 122) + 15 ;
hence not 19 divides 2333 by NAT_4:9; :: thesis:
2333 = (23 * 101) + 10 ;
hence not 23 divides 2333 by NAT_4:9; :: thesis:
2333 = (29 * 80) + 13 ;
hence not 29 divides 2333 by NAT_4:9; :: thesis:
2333 = (31 * 75) + 8 ;
hence not 31 divides 2333 by NAT_4:9; :: thesis:
2333 = (37 * 63) + 2 ;
hence not 37 divides 2333 by NAT_4:9; :: thesis:
2333 = (41 * 56) + 37 ;
hence not 41 divides 2333 by NAT_4:9; :: thesis:
2333 = (43 * 54) + 11 ;
hence not 43 divides 2333 by NAT_4:9; :: thesis:
2333 = (47 * 49) + 30 ;
hence not 47 divides 2333 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2333 & n is prime holds
not n divides 2333 by XPRIMET1:30;
hence 2333 is prime by NAT_4:14; :: thesis: