9041 = (2 * 4520) + 1 ;
hence not 2 divides 9041 by NAT_4:9; :: thesis:
9041 = (3 * 3013) + 2 ;
hence not 3 divides 9041 by NAT_4:9; :: thesis:
9041 = (5 * 1808) + 1 ;
hence not 5 divides 9041 by NAT_4:9; :: thesis:
9041 = (7 * 1291) + 4 ;
hence not 7 divides 9041 by NAT_4:9; :: thesis:
9041 = (11 * 821) + 10 ;
hence not 11 divides 9041 by NAT_4:9; :: thesis:
9041 = (13 * 695) + 6 ;
hence not 13 divides 9041 by NAT_4:9; :: thesis:
9041 = (17 * 531) + 14 ;
hence not 17 divides 9041 by NAT_4:9; :: thesis:
9041 = (19 * 475) + 16 ;
hence not 19 divides 9041 by NAT_4:9; :: thesis:
9041 = (23 * 393) + 2 ;
hence not 23 divides 9041 by NAT_4:9; :: thesis:
9041 = (29 * 311) + 22 ;
hence not 29 divides 9041 by NAT_4:9; :: thesis:
9041 = (31 * 291) + 20 ;
hence not 31 divides 9041 by NAT_4:9; :: thesis:
9041 = (37 * 244) + 13 ;
hence not 37 divides 9041 by NAT_4:9; :: thesis:
9041 = (41 * 220) + 21 ;
hence not 41 divides 9041 by NAT_4:9; :: thesis:
9041 = (43 * 210) + 11 ;
hence not 43 divides 9041 by NAT_4:9; :: thesis:
9041 = (47 * 192) + 17 ;
hence not 47 divides 9041 by NAT_4:9; :: thesis:
9041 = (53 * 170) + 31 ;
hence not 53 divides 9041 by NAT_4:9; :: thesis:
9041 = (59 * 153) + 14 ;
hence not 59 divides 9041 by NAT_4:9; :: thesis:
9041 = (61 * 148) + 13 ;
hence not 61 divides 9041 by NAT_4:9; :: thesis:
9041 = (67 * 134) + 63 ;
hence not 67 divides 9041 by NAT_4:9; :: thesis:
9041 = (71 * 127) + 24 ;
hence not 71 divides 9041 by NAT_4:9; :: thesis:
9041 = (73 * 123) + 62 ;
hence not 73 divides 9041 by NAT_4:9; :: thesis:
9041 = (79 * 114) + 35 ;
hence not 79 divides 9041 by NAT_4:9; :: thesis:
9041 = (83 * 108) + 77 ;
hence not 83 divides 9041 by NAT_4:9; :: thesis:
9041 = (89 * 101) + 52 ;
hence not 89 divides 9041 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 9041 & n is prime holds
not n divides 9041 by XPRIMET1:48;
hence 9041 is prime by NAT_4:14; :: thesis: