5323 = (2 * 2661) + 1 ;
hence not 2 divides 5323 by NAT_4:9; :: thesis:
5323 = (3 * 1774) + 1 ;
hence not 3 divides 5323 by NAT_4:9; :: thesis:
5323 = (5 * 1064) + 3 ;
hence not 5 divides 5323 by NAT_4:9; :: thesis:
5323 = (7 * 760) + 3 ;
hence not 7 divides 5323 by NAT_4:9; :: thesis:
5323 = (11 * 483) + 10 ;
hence not 11 divides 5323 by NAT_4:9; :: thesis:
5323 = (13 * 409) + 6 ;
hence not 13 divides 5323 by NAT_4:9; :: thesis:
5323 = (17 * 313) + 2 ;
hence not 17 divides 5323 by NAT_4:9; :: thesis:
5323 = (19 * 280) + 3 ;
hence not 19 divides 5323 by NAT_4:9; :: thesis:
5323 = (23 * 231) + 10 ;
hence not 23 divides 5323 by NAT_4:9; :: thesis:
5323 = (29 * 183) + 16 ;
hence not 29 divides 5323 by NAT_4:9; :: thesis:
5323 = (31 * 171) + 22 ;
hence not 31 divides 5323 by NAT_4:9; :: thesis:
5323 = (37 * 143) + 32 ;
hence not 37 divides 5323 by NAT_4:9; :: thesis:
5323 = (41 * 129) + 34 ;
hence not 41 divides 5323 by NAT_4:9; :: thesis:
5323 = (43 * 123) + 34 ;
hence not 43 divides 5323 by NAT_4:9; :: thesis:
5323 = (47 * 113) + 12 ;
hence not 47 divides 5323 by NAT_4:9; :: thesis:
5323 = (53 * 100) + 23 ;
hence not 53 divides 5323 by NAT_4:9; :: thesis:
5323 = (59 * 90) + 13 ;
hence not 59 divides 5323 by NAT_4:9; :: thesis:
5323 = (61 * 87) + 16 ;
hence not 61 divides 5323 by NAT_4:9; :: thesis:
5323 = (67 * 79) + 30 ;
hence not 67 divides 5323 by NAT_4:9; :: thesis:
5323 = (71 * 74) + 69 ;
hence not 71 divides 5323 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 5323 & n is prime holds
not n divides 5323 by XPRIMET1:40;
hence 5323 is prime by NAT_4:14; :: thesis: