2617 = (2 * 1308) + 1 ;
hence not 2 divides 2617 by NAT_4:9; :: thesis:
2617 = (3 * 872) + 1 ;
hence not 3 divides 2617 by NAT_4:9; :: thesis:
2617 = (5 * 523) + 2 ;
hence not 5 divides 2617 by NAT_4:9; :: thesis:
2617 = (7 * 373) + 6 ;
hence not 7 divides 2617 by NAT_4:9; :: thesis:
2617 = (11 * 237) + 10 ;
hence not 11 divides 2617 by NAT_4:9; :: thesis:
2617 = (13 * 201) + 4 ;
hence not 13 divides 2617 by NAT_4:9; :: thesis:
2617 = (17 * 153) + 16 ;
hence not 17 divides 2617 by NAT_4:9; :: thesis:
2617 = (19 * 137) + 14 ;
hence not 19 divides 2617 by NAT_4:9; :: thesis:
2617 = (23 * 113) + 18 ;
hence not 23 divides 2617 by NAT_4:9; :: thesis:
2617 = (29 * 90) + 7 ;
hence not 29 divides 2617 by NAT_4:9; :: thesis:
2617 = (31 * 84) + 13 ;
hence not 31 divides 2617 by NAT_4:9; :: thesis:
2617 = (37 * 70) + 27 ;
hence not 37 divides 2617 by NAT_4:9; :: thesis:
2617 = (41 * 63) + 34 ;
hence not 41 divides 2617 by NAT_4:9; :: thesis:
2617 = (43 * 60) + 37 ;
hence not 43 divides 2617 by NAT_4:9; :: thesis:
2617 = (47 * 55) + 32 ;
hence not 47 divides 2617 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2617 & n is prime holds
not n divides 2617 by XPRIMET1:30;
hence 2617 is prime by NAT_4:14; :: thesis: