2659 = (2 * 1329) + 1 ;
hence not 2 divides 2659 by NAT_4:9; :: thesis:
2659 = (3 * 886) + 1 ;
hence not 3 divides 2659 by NAT_4:9; :: thesis:
2659 = (5 * 531) + 4 ;
hence not 5 divides 2659 by NAT_4:9; :: thesis:
2659 = (7 * 379) + 6 ;
hence not 7 divides 2659 by NAT_4:9; :: thesis:
2659 = (11 * 241) + 8 ;
hence not 11 divides 2659 by NAT_4:9; :: thesis:
2659 = (13 * 204) + 7 ;
hence not 13 divides 2659 by NAT_4:9; :: thesis:
2659 = (17 * 156) + 7 ;
hence not 17 divides 2659 by NAT_4:9; :: thesis:
2659 = (19 * 139) + 18 ;
hence not 19 divides 2659 by NAT_4:9; :: thesis:
2659 = (23 * 115) + 14 ;
hence not 23 divides 2659 by NAT_4:9; :: thesis:
2659 = (29 * 91) + 20 ;
hence not 29 divides 2659 by NAT_4:9; :: thesis:
2659 = (31 * 85) + 24 ;
hence not 31 divides 2659 by NAT_4:9; :: thesis:
2659 = (37 * 71) + 32 ;
hence not 37 divides 2659 by NAT_4:9; :: thesis:
2659 = (41 * 64) + 35 ;
hence not 41 divides 2659 by NAT_4:9; :: thesis:
2659 = (43 * 61) + 36 ;
hence not 43 divides 2659 by NAT_4:9; :: thesis:
2659 = (47 * 56) + 27 ;
hence not 47 divides 2659 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 2659 & n is prime holds
not n divides 2659 by XPRIMET1:30;
hence 2659 is prime by NAT_4:14; :: thesis: