8581 = (2 * 4290) + 1 ;
hence not 2 divides 8581 by NAT_4:9; :: thesis:
8581 = (3 * 2860) + 1 ;
hence not 3 divides 8581 by NAT_4:9; :: thesis:
8581 = (5 * 1716) + 1 ;
hence not 5 divides 8581 by NAT_4:9; :: thesis:
8581 = (7 * 1225) + 6 ;
hence not 7 divides 8581 by NAT_4:9; :: thesis:
8581 = (11 * 780) + 1 ;
hence not 11 divides 8581 by NAT_4:9; :: thesis:
8581 = (13 * 660) + 1 ;
hence not 13 divides 8581 by NAT_4:9; :: thesis:
8581 = (17 * 504) + 13 ;
hence not 17 divides 8581 by NAT_4:9; :: thesis:
8581 = (19 * 451) + 12 ;
hence not 19 divides 8581 by NAT_4:9; :: thesis:
8581 = (23 * 373) + 2 ;
hence not 23 divides 8581 by NAT_4:9; :: thesis:
8581 = (29 * 295) + 26 ;
hence not 29 divides 8581 by NAT_4:9; :: thesis:
8581 = (31 * 276) + 25 ;
hence not 31 divides 8581 by NAT_4:9; :: thesis:
8581 = (37 * 231) + 34 ;
hence not 37 divides 8581 by NAT_4:9; :: thesis:
8581 = (41 * 209) + 12 ;
hence not 41 divides 8581 by NAT_4:9; :: thesis:
8581 = (43 * 199) + 24 ;
hence not 43 divides 8581 by NAT_4:9; :: thesis:
8581 = (47 * 182) + 27 ;
hence not 47 divides 8581 by NAT_4:9; :: thesis:
8581 = (53 * 161) + 48 ;
hence not 53 divides 8581 by NAT_4:9; :: thesis:
8581 = (59 * 145) + 26 ;
hence not 59 divides 8581 by NAT_4:9; :: thesis:
8581 = (61 * 140) + 41 ;
hence not 61 divides 8581 by NAT_4:9; :: thesis:
8581 = (67 * 128) + 5 ;
hence not 67 divides 8581 by NAT_4:9; :: thesis:
8581 = (71 * 120) + 61 ;
hence not 71 divides 8581 by NAT_4:9; :: thesis:
8581 = (73 * 117) + 40 ;
hence not 73 divides 8581 by NAT_4:9; :: thesis:
8581 = (79 * 108) + 49 ;
hence not 79 divides 8581 by NAT_4:9; :: thesis:
8581 = (83 * 103) + 32 ;
hence not 83 divides 8581 by NAT_4:9; :: thesis:
8581 = (89 * 96) + 37 ;
hence not 89 divides 8581 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8581 & n is prime holds
not n divides 8581 by XPRIMET1:48;
hence 8581 is prime by NAT_4:14; :: thesis: