8443 = (2 * 4221) + 1 ;
hence not 2 divides 8443 by NAT_4:9; :: thesis:
8443 = (3 * 2814) + 1 ;
hence not 3 divides 8443 by NAT_4:9; :: thesis:
8443 = (5 * 1688) + 3 ;
hence not 5 divides 8443 by NAT_4:9; :: thesis:
8443 = (7 * 1206) + 1 ;
hence not 7 divides 8443 by NAT_4:9; :: thesis:
8443 = (11 * 767) + 6 ;
hence not 11 divides 8443 by NAT_4:9; :: thesis:
8443 = (13 * 649) + 6 ;
hence not 13 divides 8443 by NAT_4:9; :: thesis:
8443 = (17 * 496) + 11 ;
hence not 17 divides 8443 by NAT_4:9; :: thesis:
8443 = (19 * 444) + 7 ;
hence not 19 divides 8443 by NAT_4:9; :: thesis:
8443 = (23 * 367) + 2 ;
hence not 23 divides 8443 by NAT_4:9; :: thesis:
8443 = (29 * 291) + 4 ;
hence not 29 divides 8443 by NAT_4:9; :: thesis:
8443 = (31 * 272) + 11 ;
hence not 31 divides 8443 by NAT_4:9; :: thesis:
8443 = (37 * 228) + 7 ;
hence not 37 divides 8443 by NAT_4:9; :: thesis:
8443 = (41 * 205) + 38 ;
hence not 41 divides 8443 by NAT_4:9; :: thesis:
8443 = (43 * 196) + 15 ;
hence not 43 divides 8443 by NAT_4:9; :: thesis:
8443 = (47 * 179) + 30 ;
hence not 47 divides 8443 by NAT_4:9; :: thesis:
8443 = (53 * 159) + 16 ;
hence not 53 divides 8443 by NAT_4:9; :: thesis:
8443 = (59 * 143) + 6 ;
hence not 59 divides 8443 by NAT_4:9; :: thesis:
8443 = (61 * 138) + 25 ;
hence not 61 divides 8443 by NAT_4:9; :: thesis:
8443 = (67 * 126) + 1 ;
hence not 67 divides 8443 by NAT_4:9; :: thesis:
8443 = (71 * 118) + 65 ;
hence not 71 divides 8443 by NAT_4:9; :: thesis:
8443 = (73 * 115) + 48 ;
hence not 73 divides 8443 by NAT_4:9; :: thesis:
8443 = (79 * 106) + 69 ;
hence not 79 divides 8443 by NAT_4:9; :: thesis:
8443 = (83 * 101) + 60 ;
hence not 83 divides 8443 by NAT_4:9; :: thesis:
8443 = (89 * 94) + 77 ;
hence not 89 divides 8443 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8443 & n is prime holds
not n divides 8443 by XPRIMET1:48;
hence 8443 is prime by NAT_4:14; :: thesis: