9001 = (2 * 4500) + 1 ;
hence not 2 divides 9001 by NAT_4:9; :: thesis:
9001 = (3 * 3000) + 1 ;
hence not 3 divides 9001 by NAT_4:9; :: thesis:
9001 = (5 * 1800) + 1 ;
hence not 5 divides 9001 by NAT_4:9; :: thesis:
9001 = (7 * 1285) + 6 ;
hence not 7 divides 9001 by NAT_4:9; :: thesis:
9001 = (11 * 818) + 3 ;
hence not 11 divides 9001 by NAT_4:9; :: thesis:
9001 = (13 * 692) + 5 ;
hence not 13 divides 9001 by NAT_4:9; :: thesis:
9001 = (17 * 529) + 8 ;
hence not 17 divides 9001 by NAT_4:9; :: thesis:
9001 = (19 * 473) + 14 ;
hence not 19 divides 9001 by NAT_4:9; :: thesis:
9001 = (23 * 391) + 8 ;
hence not 23 divides 9001 by NAT_4:9; :: thesis:
9001 = (29 * 310) + 11 ;
hence not 29 divides 9001 by NAT_4:9; :: thesis:
9001 = (31 * 290) + 11 ;
hence not 31 divides 9001 by NAT_4:9; :: thesis:
9001 = (37 * 243) + 10 ;
hence not 37 divides 9001 by NAT_4:9; :: thesis:
9001 = (41 * 219) + 22 ;
hence not 41 divides 9001 by NAT_4:9; :: thesis:
9001 = (43 * 209) + 14 ;
hence not 43 divides 9001 by NAT_4:9; :: thesis:
9001 = (47 * 191) + 24 ;
hence not 47 divides 9001 by NAT_4:9; :: thesis:
9001 = (53 * 169) + 44 ;
hence not 53 divides 9001 by NAT_4:9; :: thesis:
9001 = (59 * 152) + 33 ;
hence not 59 divides 9001 by NAT_4:9; :: thesis:
9001 = (61 * 147) + 34 ;
hence not 61 divides 9001 by NAT_4:9; :: thesis:
9001 = (67 * 134) + 23 ;
hence not 67 divides 9001 by NAT_4:9; :: thesis:
9001 = (71 * 126) + 55 ;
hence not 71 divides 9001 by NAT_4:9; :: thesis:
9001 = (73 * 123) + 22 ;
hence not 73 divides 9001 by NAT_4:9; :: thesis:
9001 = (79 * 113) + 74 ;
hence not 79 divides 9001 by NAT_4:9; :: thesis:
9001 = (83 * 108) + 37 ;
hence not 83 divides 9001 by NAT_4:9; :: thesis:
9001 = (89 * 101) + 12 ;
hence not 89 divides 9001 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9001 & n is prime holds
not n divides 9001 by XPRIMET1:48;
hence 9001 is prime by NAT_4:14; :: thesis: