3217 = (2 * 1608) + 1 ;
hence not 2 divides 3217 by NAT_4:9; :: thesis:
3217 = (3 * 1072) + 1 ;
hence not 3 divides 3217 by NAT_4:9; :: thesis:
3217 = (5 * 643) + 2 ;
hence not 5 divides 3217 by NAT_4:9; :: thesis:
3217 = (7 * 459) + 4 ;
hence not 7 divides 3217 by NAT_4:9; :: thesis:
3217 = (11 * 292) + 5 ;
hence not 11 divides 3217 by NAT_4:9; :: thesis:
3217 = (13 * 247) + 6 ;
hence not 13 divides 3217 by NAT_4:9; :: thesis:
3217 = (17 * 189) + 4 ;
hence not 17 divides 3217 by NAT_4:9; :: thesis:
3217 = (19 * 169) + 6 ;
hence not 19 divides 3217 by NAT_4:9; :: thesis:
3217 = (23 * 139) + 20 ;
hence not 23 divides 3217 by NAT_4:9; :: thesis:
3217 = (29 * 110) + 27 ;
hence not 29 divides 3217 by NAT_4:9; :: thesis:
3217 = (31 * 103) + 24 ;
hence not 31 divides 3217 by NAT_4:9; :: thesis:
3217 = (37 * 86) + 35 ;
hence not 37 divides 3217 by NAT_4:9; :: thesis:
3217 = (41 * 78) + 19 ;
hence not 41 divides 3217 by NAT_4:9; :: thesis:
3217 = (43 * 74) + 35 ;
hence not 43 divides 3217 by NAT_4:9; :: thesis:
3217 = (47 * 68) + 21 ;
hence not 47 divides 3217 by NAT_4:9; :: thesis:
3217 = (53 * 60) + 37 ;
hence not 53 divides 3217 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3217 & n is prime holds
not n divides 3217 by XPRIMET1:32;
hence 3217 is prime by NAT_4:14; :: thesis: