9013 = (2 * 4506) + 1 ;
hence not 2 divides 9013 by NAT_4:9; :: thesis:
9013 = (3 * 3004) + 1 ;
hence not 3 divides 9013 by NAT_4:9; :: thesis:
9013 = (5 * 1802) + 3 ;
hence not 5 divides 9013 by NAT_4:9; :: thesis:
9013 = (7 * 1287) + 4 ;
hence not 7 divides 9013 by NAT_4:9; :: thesis:
9013 = (11 * 819) + 4 ;
hence not 11 divides 9013 by NAT_4:9; :: thesis:
9013 = (13 * 693) + 4 ;
hence not 13 divides 9013 by NAT_4:9; :: thesis:
9013 = (17 * 530) + 3 ;
hence not 17 divides 9013 by NAT_4:9; :: thesis:
9013 = (19 * 474) + 7 ;
hence not 19 divides 9013 by NAT_4:9; :: thesis:
9013 = (23 * 391) + 20 ;
hence not 23 divides 9013 by NAT_4:9; :: thesis:
9013 = (29 * 310) + 23 ;
hence not 29 divides 9013 by NAT_4:9; :: thesis:
9013 = (31 * 290) + 23 ;
hence not 31 divides 9013 by NAT_4:9; :: thesis:
9013 = (37 * 243) + 22 ;
hence not 37 divides 9013 by NAT_4:9; :: thesis:
9013 = (41 * 219) + 34 ;
hence not 41 divides 9013 by NAT_4:9; :: thesis:
9013 = (43 * 209) + 26 ;
hence not 43 divides 9013 by NAT_4:9; :: thesis:
9013 = (47 * 191) + 36 ;
hence not 47 divides 9013 by NAT_4:9; :: thesis:
9013 = (53 * 170) + 3 ;
hence not 53 divides 9013 by NAT_4:9; :: thesis:
9013 = (59 * 152) + 45 ;
hence not 59 divides 9013 by NAT_4:9; :: thesis:
9013 = (61 * 147) + 46 ;
hence not 61 divides 9013 by NAT_4:9; :: thesis:
9013 = (67 * 134) + 35 ;
hence not 67 divides 9013 by NAT_4:9; :: thesis:
9013 = (71 * 126) + 67 ;
hence not 71 divides 9013 by NAT_4:9; :: thesis:
9013 = (73 * 123) + 34 ;
hence not 73 divides 9013 by NAT_4:9; :: thesis:
9013 = (79 * 114) + 7 ;
hence not 79 divides 9013 by NAT_4:9; :: thesis:
9013 = (83 * 108) + 49 ;
hence not 83 divides 9013 by NAT_4:9; :: thesis:
9013 = (89 * 101) + 24 ;
hence not 89 divides 9013 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9013 & n is prime holds
not n divides 9013 by XPRIMET1:48;
hence 9013 is prime by NAT_4:14; :: thesis: