3121 = (2 * 1560) + 1 ;
hence not 2 divides 3121 by NAT_4:9; :: thesis:
3121 = (3 * 1040) + 1 ;
hence not 3 divides 3121 by NAT_4:9; :: thesis:
3121 = (5 * 624) + 1 ;
hence not 5 divides 3121 by NAT_4:9; :: thesis:
3121 = (7 * 445) + 6 ;
hence not 7 divides 3121 by NAT_4:9; :: thesis:
3121 = (11 * 283) + 8 ;
hence not 11 divides 3121 by NAT_4:9; :: thesis:
3121 = (13 * 240) + 1 ;
hence not 13 divides 3121 by NAT_4:9; :: thesis:
3121 = (17 * 183) + 10 ;
hence not 17 divides 3121 by NAT_4:9; :: thesis:
3121 = (19 * 164) + 5 ;
hence not 19 divides 3121 by NAT_4:9; :: thesis:
3121 = (23 * 135) + 16 ;
hence not 23 divides 3121 by NAT_4:9; :: thesis:
3121 = (29 * 107) + 18 ;
hence not 29 divides 3121 by NAT_4:9; :: thesis:
3121 = (31 * 100) + 21 ;
hence not 31 divides 3121 by NAT_4:9; :: thesis:
3121 = (37 * 84) + 13 ;
hence not 37 divides 3121 by NAT_4:9; :: thesis:
3121 = (41 * 76) + 5 ;
hence not 41 divides 3121 by NAT_4:9; :: thesis:
3121 = (43 * 72) + 25 ;
hence not 43 divides 3121 by NAT_4:9; :: thesis:
3121 = (47 * 66) + 19 ;
hence not 47 divides 3121 by NAT_4:9; :: thesis:
3121 = (53 * 58) + 47 ;
hence not 53 divides 3121 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 3121 & n is prime holds
not n divides 3121 by XPRIMET1:32;
hence 3121 is prime by NAT_4:14; :: thesis: