3209 = (2 * 1604) + 1 ;
hence not 2 divides 3209 by NAT_4:9; :: thesis:
3209 = (3 * 1069) + 2 ;
hence not 3 divides 3209 by NAT_4:9; :: thesis:
3209 = (5 * 641) + 4 ;
hence not 5 divides 3209 by NAT_4:9; :: thesis:
3209 = (7 * 458) + 3 ;
hence not 7 divides 3209 by NAT_4:9; :: thesis:
3209 = (11 * 291) + 8 ;
hence not 11 divides 3209 by NAT_4:9; :: thesis:
3209 = (13 * 246) + 11 ;
hence not 13 divides 3209 by NAT_4:9; :: thesis:
3209 = (17 * 188) + 13 ;
hence not 17 divides 3209 by NAT_4:9; :: thesis:
3209 = (19 * 168) + 17 ;
hence not 19 divides 3209 by NAT_4:9; :: thesis:
3209 = (23 * 139) + 12 ;
hence not 23 divides 3209 by NAT_4:9; :: thesis:
3209 = (29 * 110) + 19 ;
hence not 29 divides 3209 by NAT_4:9; :: thesis:
3209 = (31 * 103) + 16 ;
hence not 31 divides 3209 by NAT_4:9; :: thesis:
3209 = (37 * 86) + 27 ;
hence not 37 divides 3209 by NAT_4:9; :: thesis:
3209 = (41 * 78) + 11 ;
hence not 41 divides 3209 by NAT_4:9; :: thesis:
3209 = (43 * 74) + 27 ;
hence not 43 divides 3209 by NAT_4:9; :: thesis:
3209 = (47 * 68) + 13 ;
hence not 47 divides 3209 by NAT_4:9; :: thesis:
3209 = (53 * 60) + 29 ;
hence not 53 divides 3209 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3209 & n is prime holds
not n divides 3209 by XPRIMET1:32;
hence 3209 is prime by NAT_4:14; :: thesis: