2281 = (2 * 1140) + 1 ;
hence not 2 divides 2281 by NAT_4:9; :: thesis:
2281 = (3 * 760) + 1 ;
hence not 3 divides 2281 by NAT_4:9; :: thesis:
2281 = (5 * 456) + 1 ;
hence not 5 divides 2281 by NAT_4:9; :: thesis:
2281 = (7 * 325) + 6 ;
hence not 7 divides 2281 by NAT_4:9; :: thesis:
2281 = (11 * 207) + 4 ;
hence not 11 divides 2281 by NAT_4:9; :: thesis:
2281 = (13 * 175) + 6 ;
hence not 13 divides 2281 by NAT_4:9; :: thesis:
2281 = (17 * 134) + 3 ;
hence not 17 divides 2281 by NAT_4:9; :: thesis:
2281 = (19 * 120) + 1 ;
hence not 19 divides 2281 by NAT_4:9; :: thesis:
2281 = (23 * 99) + 4 ;
hence not 23 divides 2281 by NAT_4:9; :: thesis:
2281 = (29 * 78) + 19 ;
hence not 29 divides 2281 by NAT_4:9; :: thesis:
2281 = (31 * 73) + 18 ;
hence not 31 divides 2281 by NAT_4:9; :: thesis:
2281 = (37 * 61) + 24 ;
hence not 37 divides 2281 by NAT_4:9; :: thesis:
2281 = (41 * 55) + 26 ;
hence not 41 divides 2281 by NAT_4:9; :: thesis:
2281 = (43 * 53) + 2 ;
hence not 43 divides 2281 by NAT_4:9; :: thesis:
2281 = (47 * 48) + 25 ;
hence not 47 divides 2281 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2281 & n is prime holds
not n divides 2281 by XPRIMET1:30;
hence 2281 is prime by NAT_4:14; :: thesis: