8573 = (2 * 4286) + 1 ;
hence not 2 divides 8573 by NAT_4:9; :: thesis:
8573 = (3 * 2857) + 2 ;
hence not 3 divides 8573 by NAT_4:9; :: thesis:
8573 = (5 * 1714) + 3 ;
hence not 5 divides 8573 by NAT_4:9; :: thesis:
8573 = (7 * 1224) + 5 ;
hence not 7 divides 8573 by NAT_4:9; :: thesis:
8573 = (11 * 779) + 4 ;
hence not 11 divides 8573 by NAT_4:9; :: thesis:
8573 = (13 * 659) + 6 ;
hence not 13 divides 8573 by NAT_4:9; :: thesis:
8573 = (17 * 504) + 5 ;
hence not 17 divides 8573 by NAT_4:9; :: thesis:
8573 = (19 * 451) + 4 ;
hence not 19 divides 8573 by NAT_4:9; :: thesis:
8573 = (23 * 372) + 17 ;
hence not 23 divides 8573 by NAT_4:9; :: thesis:
8573 = (29 * 295) + 18 ;
hence not 29 divides 8573 by NAT_4:9; :: thesis:
8573 = (31 * 276) + 17 ;
hence not 31 divides 8573 by NAT_4:9; :: thesis:
8573 = (37 * 231) + 26 ;
hence not 37 divides 8573 by NAT_4:9; :: thesis:
8573 = (41 * 209) + 4 ;
hence not 41 divides 8573 by NAT_4:9; :: thesis:
8573 = (43 * 199) + 16 ;
hence not 43 divides 8573 by NAT_4:9; :: thesis:
8573 = (47 * 182) + 19 ;
hence not 47 divides 8573 by NAT_4:9; :: thesis:
8573 = (53 * 161) + 40 ;
hence not 53 divides 8573 by NAT_4:9; :: thesis:
8573 = (59 * 145) + 18 ;
hence not 59 divides 8573 by NAT_4:9; :: thesis:
8573 = (61 * 140) + 33 ;
hence not 61 divides 8573 by NAT_4:9; :: thesis:
8573 = (67 * 127) + 64 ;
hence not 67 divides 8573 by NAT_4:9; :: thesis:
8573 = (71 * 120) + 53 ;
hence not 71 divides 8573 by NAT_4:9; :: thesis:
8573 = (73 * 117) + 32 ;
hence not 73 divides 8573 by NAT_4:9; :: thesis:
8573 = (79 * 108) + 41 ;
hence not 79 divides 8573 by NAT_4:9; :: thesis:
8573 = (83 * 103) + 24 ;
hence not 83 divides 8573 by NAT_4:9; :: thesis:
8573 = (89 * 96) + 29 ;
hence not 89 divides 8573 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8573 & n is prime holds
not n divides 8573 by XPRIMET1:48;
hence 8573 is prime by NAT_4:14; :: thesis: