6689 = (2 * 3344) + 1 ;
hence not 2 divides 6689 by NAT_4:9; :: thesis:
6689 = (3 * 2229) + 2 ;
hence not 3 divides 6689 by NAT_4:9; :: thesis:
6689 = (5 * 1337) + 4 ;
hence not 5 divides 6689 by NAT_4:9; :: thesis:
6689 = (7 * 955) + 4 ;
hence not 7 divides 6689 by NAT_4:9; :: thesis:
6689 = (11 * 608) + 1 ;
hence not 11 divides 6689 by NAT_4:9; :: thesis:
6689 = (13 * 514) + 7 ;
hence not 13 divides 6689 by NAT_4:9; :: thesis:
6689 = (17 * 393) + 8 ;
hence not 17 divides 6689 by NAT_4:9; :: thesis:
6689 = (19 * 352) + 1 ;
hence not 19 divides 6689 by NAT_4:9; :: thesis:
6689 = (23 * 290) + 19 ;
hence not 23 divides 6689 by NAT_4:9; :: thesis:
6689 = (29 * 230) + 19 ;
hence not 29 divides 6689 by NAT_4:9; :: thesis:
6689 = (31 * 215) + 24 ;
hence not 31 divides 6689 by NAT_4:9; :: thesis:
6689 = (37 * 180) + 29 ;
hence not 37 divides 6689 by NAT_4:9; :: thesis:
6689 = (41 * 163) + 6 ;
hence not 41 divides 6689 by NAT_4:9; :: thesis:
6689 = (43 * 155) + 24 ;
hence not 43 divides 6689 by NAT_4:9; :: thesis:
6689 = (47 * 142) + 15 ;
hence not 47 divides 6689 by NAT_4:9; :: thesis:
6689 = (53 * 126) + 11 ;
hence not 53 divides 6689 by NAT_4:9; :: thesis:
6689 = (59 * 113) + 22 ;
hence not 59 divides 6689 by NAT_4:9; :: thesis:
6689 = (61 * 109) + 40 ;
hence not 61 divides 6689 by NAT_4:9; :: thesis:
6689 = (67 * 99) + 56 ;
hence not 67 divides 6689 by NAT_4:9; :: thesis:
6689 = (71 * 94) + 15 ;
hence not 71 divides 6689 by NAT_4:9; :: thesis:
6689 = (73 * 91) + 46 ;
hence not 73 divides 6689 by NAT_4:9; :: thesis:
6689 = (79 * 84) + 53 ;
hence not 79 divides 6689 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6689 & n is prime holds
not n divides 6689 by XPRIMET1:44;
hence 6689 is prime by NAT_4:14; :: thesis: