3323 = (2 * 1661) + 1 ;
hence not 2 divides 3323 by NAT_4:9; :: thesis:
3323 = (3 * 1107) + 2 ;
hence not 3 divides 3323 by NAT_4:9; :: thesis:
3323 = (5 * 664) + 3 ;
hence not 5 divides 3323 by NAT_4:9; :: thesis:
3323 = (7 * 474) + 5 ;
hence not 7 divides 3323 by NAT_4:9; :: thesis:
3323 = (11 * 302) + 1 ;
hence not 11 divides 3323 by NAT_4:9; :: thesis:
3323 = (13 * 255) + 8 ;
hence not 13 divides 3323 by NAT_4:9; :: thesis:
3323 = (17 * 195) + 8 ;
hence not 17 divides 3323 by NAT_4:9; :: thesis:
3323 = (19 * 174) + 17 ;
hence not 19 divides 3323 by NAT_4:9; :: thesis:
3323 = (23 * 144) + 11 ;
hence not 23 divides 3323 by NAT_4:9; :: thesis:
3323 = (29 * 114) + 17 ;
hence not 29 divides 3323 by NAT_4:9; :: thesis:
3323 = (31 * 107) + 6 ;
hence not 31 divides 3323 by NAT_4:9; :: thesis:
3323 = (37 * 89) + 30 ;
hence not 37 divides 3323 by NAT_4:9; :: thesis:
3323 = (41 * 81) + 2 ;
hence not 41 divides 3323 by NAT_4:9; :: thesis:
3323 = (43 * 77) + 12 ;
hence not 43 divides 3323 by NAT_4:9; :: thesis:
3323 = (47 * 70) + 33 ;
hence not 47 divides 3323 by NAT_4:9; :: thesis:
3323 = (53 * 62) + 37 ;
hence not 53 divides 3323 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3323 & n is prime holds
not n divides 3323 by XPRIMET1:32;
hence 3323 is prime by NAT_4:14; :: thesis: