3169 = (2 * 1584) + 1 ;
hence not 2 divides 3169 by NAT_4:9; :: thesis:
3169 = (3 * 1056) + 1 ;
hence not 3 divides 3169 by NAT_4:9; :: thesis:
3169 = (5 * 633) + 4 ;
hence not 5 divides 3169 by NAT_4:9; :: thesis:
3169 = (7 * 452) + 5 ;
hence not 7 divides 3169 by NAT_4:9; :: thesis:
3169 = (11 * 288) + 1 ;
hence not 11 divides 3169 by NAT_4:9; :: thesis:
3169 = (13 * 243) + 10 ;
hence not 13 divides 3169 by NAT_4:9; :: thesis:
3169 = (17 * 186) + 7 ;
hence not 17 divides 3169 by NAT_4:9; :: thesis:
3169 = (19 * 166) + 15 ;
hence not 19 divides 3169 by NAT_4:9; :: thesis:
3169 = (23 * 137) + 18 ;
hence not 23 divides 3169 by NAT_4:9; :: thesis:
3169 = (29 * 109) + 8 ;
hence not 29 divides 3169 by NAT_4:9; :: thesis:
3169 = (31 * 102) + 7 ;
hence not 31 divides 3169 by NAT_4:9; :: thesis:
3169 = (37 * 85) + 24 ;
hence not 37 divides 3169 by NAT_4:9; :: thesis:
3169 = (41 * 77) + 12 ;
hence not 41 divides 3169 by NAT_4:9; :: thesis:
3169 = (43 * 73) + 30 ;
hence not 43 divides 3169 by NAT_4:9; :: thesis:
3169 = (47 * 67) + 20 ;
hence not 47 divides 3169 by NAT_4:9; :: thesis:
3169 = (53 * 59) + 42 ;
hence not 53 divides 3169 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3169 & n is prime holds
not n divides 3169 by XPRIMET1:32;
hence 3169 is prime by NAT_4:14; :: thesis: