3109 = (2 * 1554) + 1 ;
hence not 2 divides 3109 by NAT_4:9; :: thesis:
3109 = (3 * 1036) + 1 ;
hence not 3 divides 3109 by NAT_4:9; :: thesis:
3109 = (5 * 621) + 4 ;
hence not 5 divides 3109 by NAT_4:9; :: thesis:
3109 = (7 * 444) + 1 ;
hence not 7 divides 3109 by NAT_4:9; :: thesis:
3109 = (11 * 282) + 7 ;
hence not 11 divides 3109 by NAT_4:9; :: thesis:
3109 = (13 * 239) + 2 ;
hence not 13 divides 3109 by NAT_4:9; :: thesis:
3109 = (17 * 182) + 15 ;
hence not 17 divides 3109 by NAT_4:9; :: thesis:
3109 = (19 * 163) + 12 ;
hence not 19 divides 3109 by NAT_4:9; :: thesis:
3109 = (23 * 135) + 4 ;
hence not 23 divides 3109 by NAT_4:9; :: thesis:
3109 = (29 * 107) + 6 ;
hence not 29 divides 3109 by NAT_4:9; :: thesis:
3109 = (31 * 100) + 9 ;
hence not 31 divides 3109 by NAT_4:9; :: thesis:
3109 = (37 * 84) + 1 ;
hence not 37 divides 3109 by NAT_4:9; :: thesis:
3109 = (41 * 75) + 34 ;
hence not 41 divides 3109 by NAT_4:9; :: thesis:
3109 = (43 * 72) + 13 ;
hence not 43 divides 3109 by NAT_4:9; :: thesis:
3109 = (47 * 66) + 7 ;
hence not 47 divides 3109 by NAT_4:9; :: thesis:
3109 = (53 * 58) + 35 ;
hence not 53 divides 3109 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3109 & n is prime holds
not n divides 3109 by XPRIMET1:32;
hence 3109 is prime by NAT_4:14; :: thesis: