9781 = (2 * 4890) + 1 ;
hence not 2 divides 9781 by NAT_4:9; :: thesis:
9781 = (3 * 3260) + 1 ;
hence not 3 divides 9781 by NAT_4:9; :: thesis:
9781 = (5 * 1956) + 1 ;
hence not 5 divides 9781 by NAT_4:9; :: thesis:
9781 = (7 * 1397) + 2 ;
hence not 7 divides 9781 by NAT_4:9; :: thesis:
9781 = (11 * 889) + 2 ;
hence not 11 divides 9781 by NAT_4:9; :: thesis:
9781 = (13 * 752) + 5 ;
hence not 13 divides 9781 by NAT_4:9; :: thesis:
9781 = (17 * 575) + 6 ;
hence not 17 divides 9781 by NAT_4:9; :: thesis:
9781 = (19 * 514) + 15 ;
hence not 19 divides 9781 by NAT_4:9; :: thesis:
9781 = (23 * 425) + 6 ;
hence not 23 divides 9781 by NAT_4:9; :: thesis:
9781 = (29 * 337) + 8 ;
hence not 29 divides 9781 by NAT_4:9; :: thesis:
9781 = (31 * 315) + 16 ;
hence not 31 divides 9781 by NAT_4:9; :: thesis:
9781 = (37 * 264) + 13 ;
hence not 37 divides 9781 by NAT_4:9; :: thesis:
9781 = (41 * 238) + 23 ;
hence not 41 divides 9781 by NAT_4:9; :: thesis:
9781 = (43 * 227) + 20 ;
hence not 43 divides 9781 by NAT_4:9; :: thesis:
9781 = (47 * 208) + 5 ;
hence not 47 divides 9781 by NAT_4:9; :: thesis:
9781 = (53 * 184) + 29 ;
hence not 53 divides 9781 by NAT_4:9; :: thesis:
9781 = (59 * 165) + 46 ;
hence not 59 divides 9781 by NAT_4:9; :: thesis:
9781 = (61 * 160) + 21 ;
hence not 61 divides 9781 by NAT_4:9; :: thesis:
9781 = (67 * 145) + 66 ;
hence not 67 divides 9781 by NAT_4:9; :: thesis:
9781 = (71 * 137) + 54 ;
hence not 71 divides 9781 by NAT_4:9; :: thesis:
9781 = (73 * 133) + 72 ;
hence not 73 divides 9781 by NAT_4:9; :: thesis:
9781 = (79 * 123) + 64 ;
hence not 79 divides 9781 by NAT_4:9; :: thesis:
9781 = (83 * 117) + 70 ;
hence not 83 divides 9781 by NAT_4:9; :: thesis:
9781 = (89 * 109) + 80 ;
hence not 89 divides 9781 by NAT_4:9; :: thesis:
9781 = (97 * 100) + 81 ;
hence not 97 divides 9781 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9781 & n is prime holds
not n divides 9781 by XPRIMET1:50;
hence 9781 is prime by NAT_4:14; :: thesis: