2113 = (2 * 1056) + 1 ;
hence not 2 divides 2113 by NAT_4:9; :: thesis:
2113 = (3 * 704) + 1 ;
hence not 3 divides 2113 by NAT_4:9; :: thesis:
2113 = (5 * 422) + 3 ;
hence not 5 divides 2113 by NAT_4:9; :: thesis:
2113 = (7 * 301) + 6 ;
hence not 7 divides 2113 by NAT_4:9; :: thesis:
2113 = (11 * 192) + 1 ;
hence not 11 divides 2113 by NAT_4:9; :: thesis:
2113 = (13 * 162) + 7 ;
hence not 13 divides 2113 by NAT_4:9; :: thesis:
2113 = (17 * 124) + 5 ;
hence not 17 divides 2113 by NAT_4:9; :: thesis:
2113 = (19 * 111) + 4 ;
hence not 19 divides 2113 by NAT_4:9; :: thesis:
2113 = (23 * 91) + 20 ;
hence not 23 divides 2113 by NAT_4:9; :: thesis:
2113 = (29 * 72) + 25 ;
hence not 29 divides 2113 by NAT_4:9; :: thesis:
2113 = (31 * 68) + 5 ;
hence not 31 divides 2113 by NAT_4:9; :: thesis:
2113 = (37 * 57) + 4 ;
hence not 37 divides 2113 by NAT_4:9; :: thesis:
2113 = (41 * 51) + 22 ;
hence not 41 divides 2113 by NAT_4:9; :: thesis:
2113 = (43 * 49) + 6 ;
hence not 43 divides 2113 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2113 & n is prime holds
not n divides 2113 by XPRIMET1:28;
hence 2113 is prime by NAT_4:14; :: thesis: