3469 = (2 * 1734) + 1 ;
hence not 2 divides 3469 by NAT_4:9; :: thesis:
3469 = (3 * 1156) + 1 ;
hence not 3 divides 3469 by NAT_4:9; :: thesis:
3469 = (5 * 693) + 4 ;
hence not 5 divides 3469 by NAT_4:9; :: thesis:
3469 = (7 * 495) + 4 ;
hence not 7 divides 3469 by NAT_4:9; :: thesis:
3469 = (11 * 315) + 4 ;
hence not 11 divides 3469 by NAT_4:9; :: thesis:
3469 = (13 * 266) + 11 ;
hence not 13 divides 3469 by NAT_4:9; :: thesis:
3469 = (17 * 204) + 1 ;
hence not 17 divides 3469 by NAT_4:9; :: thesis:
3469 = (19 * 182) + 11 ;
hence not 19 divides 3469 by NAT_4:9; :: thesis:
3469 = (23 * 150) + 19 ;
hence not 23 divides 3469 by NAT_4:9; :: thesis:
3469 = (29 * 119) + 18 ;
hence not 29 divides 3469 by NAT_4:9; :: thesis:
3469 = (31 * 111) + 28 ;
hence not 31 divides 3469 by NAT_4:9; :: thesis:
3469 = (37 * 93) + 28 ;
hence not 37 divides 3469 by NAT_4:9; :: thesis:
3469 = (41 * 84) + 25 ;
hence not 41 divides 3469 by NAT_4:9; :: thesis:
3469 = (43 * 80) + 29 ;
hence not 43 divides 3469 by NAT_4:9; :: thesis:
3469 = (47 * 73) + 38 ;
hence not 47 divides 3469 by NAT_4:9; :: thesis:
3469 = (53 * 65) + 24 ;
hence not 53 divides 3469 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3469 & n is prime holds
not n divides 3469 by XPRIMET1:32;
hence 3469 is prime by NAT_4:14; :: thesis: