2213 = (2 * 1106) + 1 ;
hence not 2 divides 2213 by NAT_4:9; :: thesis:
2213 = (3 * 737) + 2 ;
hence not 3 divides 2213 by NAT_4:9; :: thesis:
2213 = (5 * 442) + 3 ;
hence not 5 divides 2213 by NAT_4:9; :: thesis:
2213 = (7 * 316) + 1 ;
hence not 7 divides 2213 by NAT_4:9; :: thesis:
2213 = (11 * 201) + 2 ;
hence not 11 divides 2213 by NAT_4:9; :: thesis:
2213 = (13 * 170) + 3 ;
hence not 13 divides 2213 by NAT_4:9; :: thesis:
2213 = (17 * 130) + 3 ;
hence not 17 divides 2213 by NAT_4:9; :: thesis:
2213 = (19 * 116) + 9 ;
hence not 19 divides 2213 by NAT_4:9; :: thesis:
2213 = (23 * 96) + 5 ;
hence not 23 divides 2213 by NAT_4:9; :: thesis:
2213 = (29 * 76) + 9 ;
hence not 29 divides 2213 by NAT_4:9; :: thesis:
2213 = (31 * 71) + 12 ;
hence not 31 divides 2213 by NAT_4:9; :: thesis:
2213 = (37 * 59) + 30 ;
hence not 37 divides 2213 by NAT_4:9; :: thesis:
2213 = (41 * 53) + 40 ;
hence not 41 divides 2213 by NAT_4:9; :: thesis:
2213 = (43 * 51) + 20 ;
hence not 43 divides 2213 by NAT_4:9; :: thesis:
2213 = (47 * 47) + 4 ;
hence not 47 divides 2213 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2213 & n is prime holds
not n divides 2213 by XPRIMET1:30;
hence 2213 is prime by NAT_4:14; :: thesis: