3433 = (2 * 1716) + 1 ;
hence not 2 divides 3433 by NAT_4:9; :: thesis:
3433 = (3 * 1144) + 1 ;
hence not 3 divides 3433 by NAT_4:9; :: thesis:
3433 = (5 * 686) + 3 ;
hence not 5 divides 3433 by NAT_4:9; :: thesis:
3433 = (7 * 490) + 3 ;
hence not 7 divides 3433 by NAT_4:9; :: thesis:
3433 = (11 * 312) + 1 ;
hence not 11 divides 3433 by NAT_4:9; :: thesis:
3433 = (13 * 264) + 1 ;
hence not 13 divides 3433 by NAT_4:9; :: thesis:
3433 = (17 * 201) + 16 ;
hence not 17 divides 3433 by NAT_4:9; :: thesis:
3433 = (19 * 180) + 13 ;
hence not 19 divides 3433 by NAT_4:9; :: thesis:
3433 = (23 * 149) + 6 ;
hence not 23 divides 3433 by NAT_4:9; :: thesis:
3433 = (29 * 118) + 11 ;
hence not 29 divides 3433 by NAT_4:9; :: thesis:
3433 = (31 * 110) + 23 ;
hence not 31 divides 3433 by NAT_4:9; :: thesis:
3433 = (37 * 92) + 29 ;
hence not 37 divides 3433 by NAT_4:9; :: thesis:
3433 = (41 * 83) + 30 ;
hence not 41 divides 3433 by NAT_4:9; :: thesis:
3433 = (43 * 79) + 36 ;
hence not 43 divides 3433 by NAT_4:9; :: thesis:
3433 = (47 * 73) + 2 ;
hence not 47 divides 3433 by NAT_4:9; :: thesis:
3433 = (53 * 64) + 41 ;
hence not 53 divides 3433 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3433 & n is prime holds
not n divides 3433 by XPRIMET1:32;
hence 3433 is prime by NAT_4:14; :: thesis: