8537 = (2 * 4268) + 1 ;
hence not 2 divides 8537 by NAT_4:9; :: thesis:
8537 = (3 * 2845) + 2 ;
hence not 3 divides 8537 by NAT_4:9; :: thesis:
8537 = (5 * 1707) + 2 ;
hence not 5 divides 8537 by NAT_4:9; :: thesis:
8537 = (7 * 1219) + 4 ;
hence not 7 divides 8537 by NAT_4:9; :: thesis:
8537 = (11 * 776) + 1 ;
hence not 11 divides 8537 by NAT_4:9; :: thesis:
8537 = (13 * 656) + 9 ;
hence not 13 divides 8537 by NAT_4:9; :: thesis:
8537 = (17 * 502) + 3 ;
hence not 17 divides 8537 by NAT_4:9; :: thesis:
8537 = (19 * 449) + 6 ;
hence not 19 divides 8537 by NAT_4:9; :: thesis:
8537 = (23 * 371) + 4 ;
hence not 23 divides 8537 by NAT_4:9; :: thesis:
8537 = (29 * 294) + 11 ;
hence not 29 divides 8537 by NAT_4:9; :: thesis:
8537 = (31 * 275) + 12 ;
hence not 31 divides 8537 by NAT_4:9; :: thesis:
8537 = (37 * 230) + 27 ;
hence not 37 divides 8537 by NAT_4:9; :: thesis:
8537 = (41 * 208) + 9 ;
hence not 41 divides 8537 by NAT_4:9; :: thesis:
8537 = (43 * 198) + 23 ;
hence not 43 divides 8537 by NAT_4:9; :: thesis:
8537 = (47 * 181) + 30 ;
hence not 47 divides 8537 by NAT_4:9; :: thesis:
8537 = (53 * 161) + 4 ;
hence not 53 divides 8537 by NAT_4:9; :: thesis:
8537 = (59 * 144) + 41 ;
hence not 59 divides 8537 by NAT_4:9; :: thesis:
8537 = (61 * 139) + 58 ;
hence not 61 divides 8537 by NAT_4:9; :: thesis:
8537 = (67 * 127) + 28 ;
hence not 67 divides 8537 by NAT_4:9; :: thesis:
8537 = (71 * 120) + 17 ;
hence not 71 divides 8537 by NAT_4:9; :: thesis:
8537 = (73 * 116) + 69 ;
hence not 73 divides 8537 by NAT_4:9; :: thesis:
8537 = (79 * 108) + 5 ;
hence not 79 divides 8537 by NAT_4:9; :: thesis:
8537 = (83 * 102) + 71 ;
hence not 83 divides 8537 by NAT_4:9; :: thesis:
8537 = (89 * 95) + 82 ;
hence not 89 divides 8537 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8537 & n is prime holds
not n divides 8537 by XPRIMET1:48;
hence 8537 is prime by NAT_4:14; :: thesis: