3391 = (2 * 1695) + 1 ;
hence not 2 divides 3391 by NAT_4:9; :: thesis:
3391 = (3 * 1130) + 1 ;
hence not 3 divides 3391 by NAT_4:9; :: thesis:
3391 = (5 * 678) + 1 ;
hence not 5 divides 3391 by NAT_4:9; :: thesis:
3391 = (7 * 484) + 3 ;
hence not 7 divides 3391 by NAT_4:9; :: thesis:
3391 = (11 * 308) + 3 ;
hence not 11 divides 3391 by NAT_4:9; :: thesis:
3391 = (13 * 260) + 11 ;
hence not 13 divides 3391 by NAT_4:9; :: thesis:
3391 = (17 * 199) + 8 ;
hence not 17 divides 3391 by NAT_4:9; :: thesis:
3391 = (19 * 178) + 9 ;
hence not 19 divides 3391 by NAT_4:9; :: thesis:
3391 = (23 * 147) + 10 ;
hence not 23 divides 3391 by NAT_4:9; :: thesis:
3391 = (29 * 116) + 27 ;
hence not 29 divides 3391 by NAT_4:9; :: thesis:
3391 = (31 * 109) + 12 ;
hence not 31 divides 3391 by NAT_4:9; :: thesis:
3391 = (37 * 91) + 24 ;
hence not 37 divides 3391 by NAT_4:9; :: thesis:
3391 = (41 * 82) + 29 ;
hence not 41 divides 3391 by NAT_4:9; :: thesis:
3391 = (43 * 78) + 37 ;
hence not 43 divides 3391 by NAT_4:9; :: thesis:
3391 = (47 * 72) + 7 ;
hence not 47 divides 3391 by NAT_4:9; :: thesis:
3391 = (53 * 63) + 52 ;
hence not 53 divides 3391 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3391 & n is prime holds
not n divides 3391 by XPRIMET1:32;
hence 3391 is prime by NAT_4:14; :: thesis: