9769 = (2 * 4884) + 1 ;
hence not 2 divides 9769 by NAT_4:9; :: thesis:
9769 = (3 * 3256) + 1 ;
hence not 3 divides 9769 by NAT_4:9; :: thesis:
9769 = (5 * 1953) + 4 ;
hence not 5 divides 9769 by NAT_4:9; :: thesis:
9769 = (7 * 1395) + 4 ;
hence not 7 divides 9769 by NAT_4:9; :: thesis:
9769 = (11 * 888) + 1 ;
hence not 11 divides 9769 by NAT_4:9; :: thesis:
9769 = (13 * 751) + 6 ;
hence not 13 divides 9769 by NAT_4:9; :: thesis:
9769 = (17 * 574) + 11 ;
hence not 17 divides 9769 by NAT_4:9; :: thesis:
9769 = (19 * 514) + 3 ;
hence not 19 divides 9769 by NAT_4:9; :: thesis:
9769 = (23 * 424) + 17 ;
hence not 23 divides 9769 by NAT_4:9; :: thesis:
9769 = (29 * 336) + 25 ;
hence not 29 divides 9769 by NAT_4:9; :: thesis:
9769 = (31 * 315) + 4 ;
hence not 31 divides 9769 by NAT_4:9; :: thesis:
9769 = (37 * 264) + 1 ;
hence not 37 divides 9769 by NAT_4:9; :: thesis:
9769 = (41 * 238) + 11 ;
hence not 41 divides 9769 by NAT_4:9; :: thesis:
9769 = (43 * 227) + 8 ;
hence not 43 divides 9769 by NAT_4:9; :: thesis:
9769 = (47 * 207) + 40 ;
hence not 47 divides 9769 by NAT_4:9; :: thesis:
9769 = (53 * 184) + 17 ;
hence not 53 divides 9769 by NAT_4:9; :: thesis:
9769 = (59 * 165) + 34 ;
hence not 59 divides 9769 by NAT_4:9; :: thesis:
9769 = (61 * 160) + 9 ;
hence not 61 divides 9769 by NAT_4:9; :: thesis:
9769 = (67 * 145) + 54 ;
hence not 67 divides 9769 by NAT_4:9; :: thesis:
9769 = (71 * 137) + 42 ;
hence not 71 divides 9769 by NAT_4:9; :: thesis:
9769 = (73 * 133) + 60 ;
hence not 73 divides 9769 by NAT_4:9; :: thesis:
9769 = (79 * 123) + 52 ;
hence not 79 divides 9769 by NAT_4:9; :: thesis:
9769 = (83 * 117) + 58 ;
hence not 83 divides 9769 by NAT_4:9; :: thesis:
9769 = (89 * 109) + 68 ;
hence not 89 divides 9769 by NAT_4:9; :: thesis:
9769 = (97 * 100) + 69 ;
hence not 97 divides 9769 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9769 & n is prime holds
not n divides 9769 by XPRIMET1:50;
hence 9769 is prime by NAT_4:14; :: thesis: