9377 = (2 * 4688) + 1 ;
hence not 2 divides 9377 by NAT_4:9; :: thesis:
9377 = (3 * 3125) + 2 ;
hence not 3 divides 9377 by NAT_4:9; :: thesis:
9377 = (5 * 1875) + 2 ;
hence not 5 divides 9377 by NAT_4:9; :: thesis:
9377 = (7 * 1339) + 4 ;
hence not 7 divides 9377 by NAT_4:9; :: thesis:
9377 = (11 * 852) + 5 ;
hence not 11 divides 9377 by NAT_4:9; :: thesis:
9377 = (13 * 721) + 4 ;
hence not 13 divides 9377 by NAT_4:9; :: thesis:
9377 = (17 * 551) + 10 ;
hence not 17 divides 9377 by NAT_4:9; :: thesis:
9377 = (19 * 493) + 10 ;
hence not 19 divides 9377 by NAT_4:9; :: thesis:
9377 = (23 * 407) + 16 ;
hence not 23 divides 9377 by NAT_4:9; :: thesis:
9377 = (29 * 323) + 10 ;
hence not 29 divides 9377 by NAT_4:9; :: thesis:
9377 = (31 * 302) + 15 ;
hence not 31 divides 9377 by NAT_4:9; :: thesis:
9377 = (37 * 253) + 16 ;
hence not 37 divides 9377 by NAT_4:9; :: thesis:
9377 = (41 * 228) + 29 ;
hence not 41 divides 9377 by NAT_4:9; :: thesis:
9377 = (43 * 218) + 3 ;
hence not 43 divides 9377 by NAT_4:9; :: thesis:
9377 = (47 * 199) + 24 ;
hence not 47 divides 9377 by NAT_4:9; :: thesis:
9377 = (53 * 176) + 49 ;
hence not 53 divides 9377 by NAT_4:9; :: thesis:
9377 = (59 * 158) + 55 ;
hence not 59 divides 9377 by NAT_4:9; :: thesis:
9377 = (61 * 153) + 44 ;
hence not 61 divides 9377 by NAT_4:9; :: thesis:
9377 = (67 * 139) + 64 ;
hence not 67 divides 9377 by NAT_4:9; :: thesis:
9377 = (71 * 132) + 5 ;
hence not 71 divides 9377 by NAT_4:9; :: thesis:
9377 = (73 * 128) + 33 ;
hence not 73 divides 9377 by NAT_4:9; :: thesis:
9377 = (79 * 118) + 55 ;
hence not 79 divides 9377 by NAT_4:9; :: thesis:
9377 = (83 * 112) + 81 ;
hence not 83 divides 9377 by NAT_4:9; :: thesis:
9377 = (89 * 105) + 32 ;
hence not 89 divides 9377 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9377 & n is prime holds
not n divides 9377 by XPRIMET1:48;
hence 9377 is prime by NAT_4:14; :: thesis: