7309 = (2 * 3654) + 1 ;
hence not 2 divides 7309 by NAT_4:9; :: thesis:
7309 = (3 * 2436) + 1 ;
hence not 3 divides 7309 by NAT_4:9; :: thesis:
7309 = (5 * 1461) + 4 ;
hence not 5 divides 7309 by NAT_4:9; :: thesis:
7309 = (7 * 1044) + 1 ;
hence not 7 divides 7309 by NAT_4:9; :: thesis:
7309 = (11 * 664) + 5 ;
hence not 11 divides 7309 by NAT_4:9; :: thesis:
7309 = (13 * 562) + 3 ;
hence not 13 divides 7309 by NAT_4:9; :: thesis:
7309 = (17 * 429) + 16 ;
hence not 17 divides 7309 by NAT_4:9; :: thesis:
7309 = (19 * 384) + 13 ;
hence not 19 divides 7309 by NAT_4:9; :: thesis:
7309 = (23 * 317) + 18 ;
hence not 23 divides 7309 by NAT_4:9; :: thesis:
7309 = (29 * 252) + 1 ;
hence not 29 divides 7309 by NAT_4:9; :: thesis:
7309 = (31 * 235) + 24 ;
hence not 31 divides 7309 by NAT_4:9; :: thesis:
7309 = (37 * 197) + 20 ;
hence not 37 divides 7309 by NAT_4:9; :: thesis:
7309 = (41 * 178) + 11 ;
hence not 41 divides 7309 by NAT_4:9; :: thesis:
7309 = (43 * 169) + 42 ;
hence not 43 divides 7309 by NAT_4:9; :: thesis:
7309 = (47 * 155) + 24 ;
hence not 47 divides 7309 by NAT_4:9; :: thesis:
7309 = (53 * 137) + 48 ;
hence not 53 divides 7309 by NAT_4:9; :: thesis:
7309 = (59 * 123) + 52 ;
hence not 59 divides 7309 by NAT_4:9; :: thesis:
7309 = (61 * 119) + 50 ;
hence not 61 divides 7309 by NAT_4:9; :: thesis:
7309 = (67 * 109) + 6 ;
hence not 67 divides 7309 by NAT_4:9; :: thesis:
7309 = (71 * 102) + 67 ;
hence not 71 divides 7309 by NAT_4:9; :: thesis:
7309 = (73 * 100) + 9 ;
hence not 73 divides 7309 by NAT_4:9; :: thesis:
7309 = (79 * 92) + 41 ;
hence not 79 divides 7309 by NAT_4:9; :: thesis:
7309 = (83 * 88) + 5 ;
hence not 83 divides 7309 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 7309 & n is prime holds
not n divides 7309 by XPRIMET1:46;
hence 7309 is prime by NAT_4:14; :: thesis: