8669 = (2 * 4334) + 1 ;
hence not 2 divides 8669 by NAT_4:9; :: thesis:
8669 = (3 * 2889) + 2 ;
hence not 3 divides 8669 by NAT_4:9; :: thesis:
8669 = (5 * 1733) + 4 ;
hence not 5 divides 8669 by NAT_4:9; :: thesis:
8669 = (7 * 1238) + 3 ;
hence not 7 divides 8669 by NAT_4:9; :: thesis:
8669 = (11 * 788) + 1 ;
hence not 11 divides 8669 by NAT_4:9; :: thesis:
8669 = (13 * 666) + 11 ;
hence not 13 divides 8669 by NAT_4:9; :: thesis:
8669 = (17 * 509) + 16 ;
hence not 17 divides 8669 by NAT_4:9; :: thesis:
8669 = (19 * 456) + 5 ;
hence not 19 divides 8669 by NAT_4:9; :: thesis:
8669 = (23 * 376) + 21 ;
hence not 23 divides 8669 by NAT_4:9; :: thesis:
8669 = (29 * 298) + 27 ;
hence not 29 divides 8669 by NAT_4:9; :: thesis:
8669 = (31 * 279) + 20 ;
hence not 31 divides 8669 by NAT_4:9; :: thesis:
8669 = (37 * 234) + 11 ;
hence not 37 divides 8669 by NAT_4:9; :: thesis:
8669 = (41 * 211) + 18 ;
hence not 41 divides 8669 by NAT_4:9; :: thesis:
8669 = (43 * 201) + 26 ;
hence not 43 divides 8669 by NAT_4:9; :: thesis:
8669 = (47 * 184) + 21 ;
hence not 47 divides 8669 by NAT_4:9; :: thesis:
8669 = (53 * 163) + 30 ;
hence not 53 divides 8669 by NAT_4:9; :: thesis:
8669 = (59 * 146) + 55 ;
hence not 59 divides 8669 by NAT_4:9; :: thesis:
8669 = (61 * 142) + 7 ;
hence not 61 divides 8669 by NAT_4:9; :: thesis:
8669 = (67 * 129) + 26 ;
hence not 67 divides 8669 by NAT_4:9; :: thesis:
8669 = (71 * 122) + 7 ;
hence not 71 divides 8669 by NAT_4:9; :: thesis:
8669 = (73 * 118) + 55 ;
hence not 73 divides 8669 by NAT_4:9; :: thesis:
8669 = (79 * 109) + 58 ;
hence not 79 divides 8669 by NAT_4:9; :: thesis:
8669 = (83 * 104) + 37 ;
hence not 83 divides 8669 by NAT_4:9; :: thesis:
8669 = (89 * 97) + 36 ;
hence not 89 divides 8669 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 8669 & n is prime holds
not n divides 8669 by XPRIMET1:48;
hence 8669 is prime by NAT_4:14; :: thesis: