8447 = (2 * 4223) + 1 ;
hence not 2 divides 8447 by NAT_4:9; :: thesis:
8447 = (3 * 2815) + 2 ;
hence not 3 divides 8447 by NAT_4:9; :: thesis:
8447 = (5 * 1689) + 2 ;
hence not 5 divides 8447 by NAT_4:9; :: thesis:
8447 = (7 * 1206) + 5 ;
hence not 7 divides 8447 by NAT_4:9; :: thesis:
8447 = (11 * 767) + 10 ;
hence not 11 divides 8447 by NAT_4:9; :: thesis:
8447 = (13 * 649) + 10 ;
hence not 13 divides 8447 by NAT_4:9; :: thesis:
8447 = (17 * 496) + 15 ;
hence not 17 divides 8447 by NAT_4:9; :: thesis:
8447 = (19 * 444) + 11 ;
hence not 19 divides 8447 by NAT_4:9; :: thesis:
8447 = (23 * 367) + 6 ;
hence not 23 divides 8447 by NAT_4:9; :: thesis:
8447 = (29 * 291) + 8 ;
hence not 29 divides 8447 by NAT_4:9; :: thesis:
8447 = (31 * 272) + 15 ;
hence not 31 divides 8447 by NAT_4:9; :: thesis:
8447 = (37 * 228) + 11 ;
hence not 37 divides 8447 by NAT_4:9; :: thesis:
8447 = (41 * 206) + 1 ;
hence not 41 divides 8447 by NAT_4:9; :: thesis:
8447 = (43 * 196) + 19 ;
hence not 43 divides 8447 by NAT_4:9; :: thesis:
8447 = (47 * 179) + 34 ;
hence not 47 divides 8447 by NAT_4:9; :: thesis:
8447 = (53 * 159) + 20 ;
hence not 53 divides 8447 by NAT_4:9; :: thesis:
8447 = (59 * 143) + 10 ;
hence not 59 divides 8447 by NAT_4:9; :: thesis:
8447 = (61 * 138) + 29 ;
hence not 61 divides 8447 by NAT_4:9; :: thesis:
8447 = (67 * 126) + 5 ;
hence not 67 divides 8447 by NAT_4:9; :: thesis:
8447 = (71 * 118) + 69 ;
hence not 71 divides 8447 by NAT_4:9; :: thesis:
8447 = (73 * 115) + 52 ;
hence not 73 divides 8447 by NAT_4:9; :: thesis:
8447 = (79 * 106) + 73 ;
hence not 79 divides 8447 by NAT_4:9; :: thesis:
8447 = (83 * 101) + 64 ;
hence not 83 divides 8447 by NAT_4:9; :: thesis:
8447 = (89 * 94) + 81 ;
hence not 89 divides 8447 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8447 & n is prime holds
not n divides 8447 by XPRIMET1:48;
hence 8447 is prime by NAT_4:14; :: thesis: