8461 = (2 * 4230) + 1 ;
hence not 2 divides 8461 by NAT_4:9; :: thesis:
8461 = (3 * 2820) + 1 ;
hence not 3 divides 8461 by NAT_4:9; :: thesis:
8461 = (5 * 1692) + 1 ;
hence not 5 divides 8461 by NAT_4:9; :: thesis:
8461 = (7 * 1208) + 5 ;
hence not 7 divides 8461 by NAT_4:9; :: thesis:
8461 = (11 * 769) + 2 ;
hence not 11 divides 8461 by NAT_4:9; :: thesis:
8461 = (13 * 650) + 11 ;
hence not 13 divides 8461 by NAT_4:9; :: thesis:
8461 = (17 * 497) + 12 ;
hence not 17 divides 8461 by NAT_4:9; :: thesis:
8461 = (19 * 445) + 6 ;
hence not 19 divides 8461 by NAT_4:9; :: thesis:
8461 = (23 * 367) + 20 ;
hence not 23 divides 8461 by NAT_4:9; :: thesis:
8461 = (29 * 291) + 22 ;
hence not 29 divides 8461 by NAT_4:9; :: thesis:
8461 = (31 * 272) + 29 ;
hence not 31 divides 8461 by NAT_4:9; :: thesis:
8461 = (37 * 228) + 25 ;
hence not 37 divides 8461 by NAT_4:9; :: thesis:
8461 = (41 * 206) + 15 ;
hence not 41 divides 8461 by NAT_4:9; :: thesis:
8461 = (43 * 196) + 33 ;
hence not 43 divides 8461 by NAT_4:9; :: thesis:
8461 = (47 * 180) + 1 ;
hence not 47 divides 8461 by NAT_4:9; :: thesis:
8461 = (53 * 159) + 34 ;
hence not 53 divides 8461 by NAT_4:9; :: thesis:
8461 = (59 * 143) + 24 ;
hence not 59 divides 8461 by NAT_4:9; :: thesis:
8461 = (61 * 138) + 43 ;
hence not 61 divides 8461 by NAT_4:9; :: thesis:
8461 = (67 * 126) + 19 ;
hence not 67 divides 8461 by NAT_4:9; :: thesis:
8461 = (71 * 119) + 12 ;
hence not 71 divides 8461 by NAT_4:9; :: thesis:
8461 = (73 * 115) + 66 ;
hence not 73 divides 8461 by NAT_4:9; :: thesis:
8461 = (79 * 107) + 8 ;
hence not 79 divides 8461 by NAT_4:9; :: thesis:
8461 = (83 * 101) + 78 ;
hence not 83 divides 8461 by NAT_4:9; :: thesis:
8461 = (89 * 95) + 6 ;
hence not 89 divides 8461 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8461 & n is prime holds
not n divides 8461 by XPRIMET1:48;
hence 8461 is prime by NAT_4:14; :: thesis: