3389 = (2 * 1694) + 1 ;
hence not 2 divides 3389 by NAT_4:9; :: thesis:
3389 = (3 * 1129) + 2 ;
hence not 3 divides 3389 by NAT_4:9; :: thesis:
3389 = (5 * 677) + 4 ;
hence not 5 divides 3389 by NAT_4:9; :: thesis:
3389 = (7 * 484) + 1 ;
hence not 7 divides 3389 by NAT_4:9; :: thesis:
3389 = (11 * 308) + 1 ;
hence not 11 divides 3389 by NAT_4:9; :: thesis:
3389 = (13 * 260) + 9 ;
hence not 13 divides 3389 by NAT_4:9; :: thesis:
3389 = (17 * 199) + 6 ;
hence not 17 divides 3389 by NAT_4:9; :: thesis:
3389 = (19 * 178) + 7 ;
hence not 19 divides 3389 by NAT_4:9; :: thesis:
3389 = (23 * 147) + 8 ;
hence not 23 divides 3389 by NAT_4:9; :: thesis:
3389 = (29 * 116) + 25 ;
hence not 29 divides 3389 by NAT_4:9; :: thesis:
3389 = (31 * 109) + 10 ;
hence not 31 divides 3389 by NAT_4:9; :: thesis:
3389 = (37 * 91) + 22 ;
hence not 37 divides 3389 by NAT_4:9; :: thesis:
3389 = (41 * 82) + 27 ;
hence not 41 divides 3389 by NAT_4:9; :: thesis:
3389 = (43 * 78) + 35 ;
hence not 43 divides 3389 by NAT_4:9; :: thesis:
3389 = (47 * 72) + 5 ;
hence not 47 divides 3389 by NAT_4:9; :: thesis:
3389 = (53 * 63) + 50 ;
hence not 53 divides 3389 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3389 & n is prime holds
not n divides 3389 by XPRIMET1:32;
hence 3389 is prime by NAT_4:14; :: thesis: