9497 = (2 * 4748) + 1 ;
hence not 2 divides 9497 by NAT_4:9; :: thesis:
9497 = (3 * 3165) + 2 ;
hence not 3 divides 9497 by NAT_4:9; :: thesis:
9497 = (5 * 1899) + 2 ;
hence not 5 divides 9497 by NAT_4:9; :: thesis:
9497 = (7 * 1356) + 5 ;
hence not 7 divides 9497 by NAT_4:9; :: thesis:
9497 = (11 * 863) + 4 ;
hence not 11 divides 9497 by NAT_4:9; :: thesis:
9497 = (13 * 730) + 7 ;
hence not 13 divides 9497 by NAT_4:9; :: thesis:
9497 = (17 * 558) + 11 ;
hence not 17 divides 9497 by NAT_4:9; :: thesis:
9497 = (19 * 499) + 16 ;
hence not 19 divides 9497 by NAT_4:9; :: thesis:
9497 = (23 * 412) + 21 ;
hence not 23 divides 9497 by NAT_4:9; :: thesis:
9497 = (29 * 327) + 14 ;
hence not 29 divides 9497 by NAT_4:9; :: thesis:
9497 = (31 * 306) + 11 ;
hence not 31 divides 9497 by NAT_4:9; :: thesis:
9497 = (37 * 256) + 25 ;
hence not 37 divides 9497 by NAT_4:9; :: thesis:
9497 = (41 * 231) + 26 ;
hence not 41 divides 9497 by NAT_4:9; :: thesis:
9497 = (43 * 220) + 37 ;
hence not 43 divides 9497 by NAT_4:9; :: thesis:
9497 = (47 * 202) + 3 ;
hence not 47 divides 9497 by NAT_4:9; :: thesis:
9497 = (53 * 179) + 10 ;
hence not 53 divides 9497 by NAT_4:9; :: thesis:
9497 = (59 * 160) + 57 ;
hence not 59 divides 9497 by NAT_4:9; :: thesis:
9497 = (61 * 155) + 42 ;
hence not 61 divides 9497 by NAT_4:9; :: thesis:
9497 = (67 * 141) + 50 ;
hence not 67 divides 9497 by NAT_4:9; :: thesis:
9497 = (71 * 133) + 54 ;
hence not 71 divides 9497 by NAT_4:9; :: thesis:
9497 = (73 * 130) + 7 ;
hence not 73 divides 9497 by NAT_4:9; :: thesis:
9497 = (79 * 120) + 17 ;
hence not 79 divides 9497 by NAT_4:9; :: thesis:
9497 = (83 * 114) + 35 ;
hence not 83 divides 9497 by NAT_4:9; :: thesis:
9497 = (89 * 106) + 63 ;
hence not 89 divides 9497 by NAT_4:9; :: thesis:
9497 = (97 * 97) + 88 ;
hence not 97 divides 9497 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9497 & n is prime holds
not n divides 9497 by XPRIMET1:50;
hence 9497 is prime by NAT_4:14; :: thesis: