9397 = (2 * 4698) + 1 ;
hence not 2 divides 9397 by NAT_4:9; :: thesis:
9397 = (3 * 3132) + 1 ;
hence not 3 divides 9397 by NAT_4:9; :: thesis:
9397 = (5 * 1879) + 2 ;
hence not 5 divides 9397 by NAT_4:9; :: thesis:
9397 = (7 * 1342) + 3 ;
hence not 7 divides 9397 by NAT_4:9; :: thesis:
9397 = (11 * 854) + 3 ;
hence not 11 divides 9397 by NAT_4:9; :: thesis:
9397 = (13 * 722) + 11 ;
hence not 13 divides 9397 by NAT_4:9; :: thesis:
9397 = (17 * 552) + 13 ;
hence not 17 divides 9397 by NAT_4:9; :: thesis:
9397 = (19 * 494) + 11 ;
hence not 19 divides 9397 by NAT_4:9; :: thesis:
9397 = (23 * 408) + 13 ;
hence not 23 divides 9397 by NAT_4:9; :: thesis:
9397 = (29 * 324) + 1 ;
hence not 29 divides 9397 by NAT_4:9; :: thesis:
9397 = (31 * 303) + 4 ;
hence not 31 divides 9397 by NAT_4:9; :: thesis:
9397 = (37 * 253) + 36 ;
hence not 37 divides 9397 by NAT_4:9; :: thesis:
9397 = (41 * 229) + 8 ;
hence not 41 divides 9397 by NAT_4:9; :: thesis:
9397 = (43 * 218) + 23 ;
hence not 43 divides 9397 by NAT_4:9; :: thesis:
9397 = (47 * 199) + 44 ;
hence not 47 divides 9397 by NAT_4:9; :: thesis:
9397 = (53 * 177) + 16 ;
hence not 53 divides 9397 by NAT_4:9; :: thesis:
9397 = (59 * 159) + 16 ;
hence not 59 divides 9397 by NAT_4:9; :: thesis:
9397 = (61 * 154) + 3 ;
hence not 61 divides 9397 by NAT_4:9; :: thesis:
9397 = (67 * 140) + 17 ;
hence not 67 divides 9397 by NAT_4:9; :: thesis:
9397 = (71 * 132) + 25 ;
hence not 71 divides 9397 by NAT_4:9; :: thesis:
9397 = (73 * 128) + 53 ;
hence not 73 divides 9397 by NAT_4:9; :: thesis:
9397 = (79 * 118) + 75 ;
hence not 79 divides 9397 by NAT_4:9; :: thesis:
9397 = (83 * 113) + 18 ;
hence not 83 divides 9397 by NAT_4:9; :: thesis:
9397 = (89 * 105) + 52 ;
hence not 89 divides 9397 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9397 & n is prime holds
not n divides 9397 by XPRIMET1:48;
hence 9397 is prime by NAT_4:14; :: thesis: