8647 = (2 * 4323) + 1 ;
hence not 2 divides 8647 by NAT_4:9; :: thesis:
8647 = (3 * 2882) + 1 ;
hence not 3 divides 8647 by NAT_4:9; :: thesis:
8647 = (5 * 1729) + 2 ;
hence not 5 divides 8647 by NAT_4:9; :: thesis:
8647 = (7 * 1235) + 2 ;
hence not 7 divides 8647 by NAT_4:9; :: thesis:
8647 = (11 * 786) + 1 ;
hence not 11 divides 8647 by NAT_4:9; :: thesis:
8647 = (13 * 665) + 2 ;
hence not 13 divides 8647 by NAT_4:9; :: thesis:
8647 = (17 * 508) + 11 ;
hence not 17 divides 8647 by NAT_4:9; :: thesis:
8647 = (19 * 455) + 2 ;
hence not 19 divides 8647 by NAT_4:9; :: thesis:
8647 = (23 * 375) + 22 ;
hence not 23 divides 8647 by NAT_4:9; :: thesis:
8647 = (29 * 298) + 5 ;
hence not 29 divides 8647 by NAT_4:9; :: thesis:
8647 = (31 * 278) + 29 ;
hence not 31 divides 8647 by NAT_4:9; :: thesis:
8647 = (37 * 233) + 26 ;
hence not 37 divides 8647 by NAT_4:9; :: thesis:
8647 = (41 * 210) + 37 ;
hence not 41 divides 8647 by NAT_4:9; :: thesis:
8647 = (43 * 201) + 4 ;
hence not 43 divides 8647 by NAT_4:9; :: thesis:
8647 = (47 * 183) + 46 ;
hence not 47 divides 8647 by NAT_4:9; :: thesis:
8647 = (53 * 163) + 8 ;
hence not 53 divides 8647 by NAT_4:9; :: thesis:
8647 = (59 * 146) + 33 ;
hence not 59 divides 8647 by NAT_4:9; :: thesis:
8647 = (61 * 141) + 46 ;
hence not 61 divides 8647 by NAT_4:9; :: thesis:
8647 = (67 * 129) + 4 ;
hence not 67 divides 8647 by NAT_4:9; :: thesis:
8647 = (71 * 121) + 56 ;
hence not 71 divides 8647 by NAT_4:9; :: thesis:
8647 = (73 * 118) + 33 ;
hence not 73 divides 8647 by NAT_4:9; :: thesis:
8647 = (79 * 109) + 36 ;
hence not 79 divides 8647 by NAT_4:9; :: thesis:
8647 = (83 * 104) + 15 ;
hence not 83 divides 8647 by NAT_4:9; :: thesis:
8647 = (89 * 97) + 14 ;
hence not 89 divides 8647 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8647 & n is prime holds
not n divides 8647 by XPRIMET1:48;
hence 8647 is prime by NAT_4:14; :: thesis: