8527 = (2 * 4263) + 1 ;
hence not 2 divides 8527 by NAT_4:9; :: thesis:
8527 = (3 * 2842) + 1 ;
hence not 3 divides 8527 by NAT_4:9; :: thesis:
8527 = (5 * 1705) + 2 ;
hence not 5 divides 8527 by NAT_4:9; :: thesis:
8527 = (7 * 1218) + 1 ;
hence not 7 divides 8527 by NAT_4:9; :: thesis:
8527 = (11 * 775) + 2 ;
hence not 11 divides 8527 by NAT_4:9; :: thesis:
8527 = (13 * 655) + 12 ;
hence not 13 divides 8527 by NAT_4:9; :: thesis:
8527 = (17 * 501) + 10 ;
hence not 17 divides 8527 by NAT_4:9; :: thesis:
8527 = (19 * 448) + 15 ;
hence not 19 divides 8527 by NAT_4:9; :: thesis:
8527 = (23 * 370) + 17 ;
hence not 23 divides 8527 by NAT_4:9; :: thesis:
8527 = (29 * 294) + 1 ;
hence not 29 divides 8527 by NAT_4:9; :: thesis:
8527 = (31 * 275) + 2 ;
hence not 31 divides 8527 by NAT_4:9; :: thesis:
8527 = (37 * 230) + 17 ;
hence not 37 divides 8527 by NAT_4:9; :: thesis:
8527 = (41 * 207) + 40 ;
hence not 41 divides 8527 by NAT_4:9; :: thesis:
8527 = (43 * 198) + 13 ;
hence not 43 divides 8527 by NAT_4:9; :: thesis:
8527 = (47 * 181) + 20 ;
hence not 47 divides 8527 by NAT_4:9; :: thesis:
8527 = (53 * 160) + 47 ;
hence not 53 divides 8527 by NAT_4:9; :: thesis:
8527 = (59 * 144) + 31 ;
hence not 59 divides 8527 by NAT_4:9; :: thesis:
8527 = (61 * 139) + 48 ;
hence not 61 divides 8527 by NAT_4:9; :: thesis:
8527 = (67 * 127) + 18 ;
hence not 67 divides 8527 by NAT_4:9; :: thesis:
8527 = (71 * 120) + 7 ;
hence not 71 divides 8527 by NAT_4:9; :: thesis:
8527 = (73 * 116) + 59 ;
hence not 73 divides 8527 by NAT_4:9; :: thesis:
8527 = (79 * 107) + 74 ;
hence not 79 divides 8527 by NAT_4:9; :: thesis:
8527 = (83 * 102) + 61 ;
hence not 83 divides 8527 by NAT_4:9; :: thesis:
8527 = (89 * 95) + 72 ;
hence not 89 divides 8527 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8527 & n is prime holds
not n divides 8527 by XPRIMET1:48;
hence 8527 is prime by NAT_4:14; :: thesis: