6569 = (2 * 3284) + 1 ;
hence not 2 divides 6569 by NAT_4:9; :: thesis:
6569 = (3 * 2189) + 2 ;
hence not 3 divides 6569 by NAT_4:9; :: thesis:
6569 = (5 * 1313) + 4 ;
hence not 5 divides 6569 by NAT_4:9; :: thesis:
6569 = (7 * 938) + 3 ;
hence not 7 divides 6569 by NAT_4:9; :: thesis:
6569 = (11 * 597) + 2 ;
hence not 11 divides 6569 by NAT_4:9; :: thesis:
6569 = (13 * 505) + 4 ;
hence not 13 divides 6569 by NAT_4:9; :: thesis:
6569 = (17 * 386) + 7 ;
hence not 17 divides 6569 by NAT_4:9; :: thesis:
6569 = (19 * 345) + 14 ;
hence not 19 divides 6569 by NAT_4:9; :: thesis:
6569 = (23 * 285) + 14 ;
hence not 23 divides 6569 by NAT_4:9; :: thesis:
6569 = (29 * 226) + 15 ;
hence not 29 divides 6569 by NAT_4:9; :: thesis:
6569 = (31 * 211) + 28 ;
hence not 31 divides 6569 by NAT_4:9; :: thesis:
6569 = (37 * 177) + 20 ;
hence not 37 divides 6569 by NAT_4:9; :: thesis:
6569 = (41 * 160) + 9 ;
hence not 41 divides 6569 by NAT_4:9; :: thesis:
6569 = (43 * 152) + 33 ;
hence not 43 divides 6569 by NAT_4:9; :: thesis:
6569 = (47 * 139) + 36 ;
hence not 47 divides 6569 by NAT_4:9; :: thesis:
6569 = (53 * 123) + 50 ;
hence not 53 divides 6569 by NAT_4:9; :: thesis:
6569 = (59 * 111) + 20 ;
hence not 59 divides 6569 by NAT_4:9; :: thesis:
6569 = (61 * 107) + 42 ;
hence not 61 divides 6569 by NAT_4:9; :: thesis:
6569 = (67 * 98) + 3 ;
hence not 67 divides 6569 by NAT_4:9; :: thesis:
6569 = (71 * 92) + 37 ;
hence not 71 divides 6569 by NAT_4:9; :: thesis:
6569 = (73 * 89) + 72 ;
hence not 73 divides 6569 by NAT_4:9; :: thesis:
6569 = (79 * 83) + 12 ;
hence not 79 divides 6569 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6569 & n is prime holds
not n divides 6569 by XPRIMET1:44;
hence 6569 is prime by NAT_4:14; :: thesis: