8369 = (2 * 4184) + 1 ;
hence not 2 divides 8369 by NAT_4:9; :: thesis:
8369 = (3 * 2789) + 2 ;
hence not 3 divides 8369 by NAT_4:9; :: thesis:
8369 = (5 * 1673) + 4 ;
hence not 5 divides 8369 by NAT_4:9; :: thesis:
8369 = (7 * 1195) + 4 ;
hence not 7 divides 8369 by NAT_4:9; :: thesis:
8369 = (11 * 760) + 9 ;
hence not 11 divides 8369 by NAT_4:9; :: thesis:
8369 = (13 * 643) + 10 ;
hence not 13 divides 8369 by NAT_4:9; :: thesis:
8369 = (17 * 492) + 5 ;
hence not 17 divides 8369 by NAT_4:9; :: thesis:
8369 = (19 * 440) + 9 ;
hence not 19 divides 8369 by NAT_4:9; :: thesis:
8369 = (23 * 363) + 20 ;
hence not 23 divides 8369 by NAT_4:9; :: thesis:
8369 = (29 * 288) + 17 ;
hence not 29 divides 8369 by NAT_4:9; :: thesis:
8369 = (31 * 269) + 30 ;
hence not 31 divides 8369 by NAT_4:9; :: thesis:
8369 = (37 * 226) + 7 ;
hence not 37 divides 8369 by NAT_4:9; :: thesis:
8369 = (41 * 204) + 5 ;
hence not 41 divides 8369 by NAT_4:9; :: thesis:
8369 = (43 * 194) + 27 ;
hence not 43 divides 8369 by NAT_4:9; :: thesis:
8369 = (47 * 178) + 3 ;
hence not 47 divides 8369 by NAT_4:9; :: thesis:
8369 = (53 * 157) + 48 ;
hence not 53 divides 8369 by NAT_4:9; :: thesis:
8369 = (59 * 141) + 50 ;
hence not 59 divides 8369 by NAT_4:9; :: thesis:
8369 = (61 * 137) + 12 ;
hence not 61 divides 8369 by NAT_4:9; :: thesis:
8369 = (67 * 124) + 61 ;
hence not 67 divides 8369 by NAT_4:9; :: thesis:
8369 = (71 * 117) + 62 ;
hence not 71 divides 8369 by NAT_4:9; :: thesis:
8369 = (73 * 114) + 47 ;
hence not 73 divides 8369 by NAT_4:9; :: thesis:
8369 = (79 * 105) + 74 ;
hence not 79 divides 8369 by NAT_4:9; :: thesis:
8369 = (83 * 100) + 69 ;
hence not 83 divides 8369 by NAT_4:9; :: thesis:
8369 = (89 * 94) + 3 ;
hence not 89 divides 8369 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8369 & n is prime holds
not n divides 8369 by XPRIMET1:48;
hence 8369 is prime by NAT_4:14; :: thesis: