8243 = (2 * 4121) + 1 ;
hence not 2 divides 8243 by NAT_4:9; :: thesis:
8243 = (3 * 2747) + 2 ;
hence not 3 divides 8243 by NAT_4:9; :: thesis:
8243 = (5 * 1648) + 3 ;
hence not 5 divides 8243 by NAT_4:9; :: thesis:
8243 = (7 * 1177) + 4 ;
hence not 7 divides 8243 by NAT_4:9; :: thesis:
8243 = (11 * 749) + 4 ;
hence not 11 divides 8243 by NAT_4:9; :: thesis:
8243 = (13 * 634) + 1 ;
hence not 13 divides 8243 by NAT_4:9; :: thesis:
8243 = (17 * 484) + 15 ;
hence not 17 divides 8243 by NAT_4:9; :: thesis:
8243 = (19 * 433) + 16 ;
hence not 19 divides 8243 by NAT_4:9; :: thesis:
8243 = (23 * 358) + 9 ;
hence not 23 divides 8243 by NAT_4:9; :: thesis:
8243 = (29 * 284) + 7 ;
hence not 29 divides 8243 by NAT_4:9; :: thesis:
8243 = (31 * 265) + 28 ;
hence not 31 divides 8243 by NAT_4:9; :: thesis:
8243 = (37 * 222) + 29 ;
hence not 37 divides 8243 by NAT_4:9; :: thesis:
8243 = (41 * 201) + 2 ;
hence not 41 divides 8243 by NAT_4:9; :: thesis:
8243 = (43 * 191) + 30 ;
hence not 43 divides 8243 by NAT_4:9; :: thesis:
8243 = (47 * 175) + 18 ;
hence not 47 divides 8243 by NAT_4:9; :: thesis:
8243 = (53 * 155) + 28 ;
hence not 53 divides 8243 by NAT_4:9; :: thesis:
8243 = (59 * 139) + 42 ;
hence not 59 divides 8243 by NAT_4:9; :: thesis:
8243 = (61 * 135) + 8 ;
hence not 61 divides 8243 by NAT_4:9; :: thesis:
8243 = (67 * 123) + 2 ;
hence not 67 divides 8243 by NAT_4:9; :: thesis:
8243 = (71 * 116) + 7 ;
hence not 71 divides 8243 by NAT_4:9; :: thesis:
8243 = (73 * 112) + 67 ;
hence not 73 divides 8243 by NAT_4:9; :: thesis:
8243 = (79 * 104) + 27 ;
hence not 79 divides 8243 by NAT_4:9; :: thesis:
8243 = (83 * 99) + 26 ;
hence not 83 divides 8243 by NAT_4:9; :: thesis:
8243 = (89 * 92) + 55 ;
hence not 89 divides 8243 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8243 & n is prime holds
not n divides 8243 by XPRIMET1:48;
hence 8243 is prime by NAT_4:14; :: thesis: