3449 = (2 * 1724) + 1 ;
hence not 2 divides 3449 by NAT_4:9; :: thesis:
3449 = (3 * 1149) + 2 ;
hence not 3 divides 3449 by NAT_4:9; :: thesis:
3449 = (5 * 689) + 4 ;
hence not 5 divides 3449 by NAT_4:9; :: thesis:
3449 = (7 * 492) + 5 ;
hence not 7 divides 3449 by NAT_4:9; :: thesis:
3449 = (11 * 313) + 6 ;
hence not 11 divides 3449 by NAT_4:9; :: thesis:
3449 = (13 * 265) + 4 ;
hence not 13 divides 3449 by NAT_4:9; :: thesis:
3449 = (17 * 202) + 15 ;
hence not 17 divides 3449 by NAT_4:9; :: thesis:
3449 = (19 * 181) + 10 ;
hence not 19 divides 3449 by NAT_4:9; :: thesis:
3449 = (23 * 149) + 22 ;
hence not 23 divides 3449 by NAT_4:9; :: thesis:
3449 = (29 * 118) + 27 ;
hence not 29 divides 3449 by NAT_4:9; :: thesis:
3449 = (31 * 111) + 8 ;
hence not 31 divides 3449 by NAT_4:9; :: thesis:
3449 = (37 * 93) + 8 ;
hence not 37 divides 3449 by NAT_4:9; :: thesis:
3449 = (41 * 84) + 5 ;
hence not 41 divides 3449 by NAT_4:9; :: thesis:
3449 = (43 * 80) + 9 ;
hence not 43 divides 3449 by NAT_4:9; :: thesis:
3449 = (47 * 73) + 18 ;
hence not 47 divides 3449 by NAT_4:9; :: thesis:
3449 = (53 * 65) + 4 ;
hence not 53 divides 3449 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3449 & n is prime holds
not n divides 3449 by XPRIMET1:32;
hence 3449 is prime by NAT_4:14; :: thesis: