6577 = (2 * 3288) + 1 ;
hence not 2 divides 6577 by NAT_4:9; :: thesis:
6577 = (3 * 2192) + 1 ;
hence not 3 divides 6577 by NAT_4:9; :: thesis:
6577 = (5 * 1315) + 2 ;
hence not 5 divides 6577 by NAT_4:9; :: thesis:
6577 = (7 * 939) + 4 ;
hence not 7 divides 6577 by NAT_4:9; :: thesis:
6577 = (11 * 597) + 10 ;
hence not 11 divides 6577 by NAT_4:9; :: thesis:
6577 = (13 * 505) + 12 ;
hence not 13 divides 6577 by NAT_4:9; :: thesis:
6577 = (17 * 386) + 15 ;
hence not 17 divides 6577 by NAT_4:9; :: thesis:
6577 = (19 * 346) + 3 ;
hence not 19 divides 6577 by NAT_4:9; :: thesis:
6577 = (23 * 285) + 22 ;
hence not 23 divides 6577 by NAT_4:9; :: thesis:
6577 = (29 * 226) + 23 ;
hence not 29 divides 6577 by NAT_4:9; :: thesis:
6577 = (31 * 212) + 5 ;
hence not 31 divides 6577 by NAT_4:9; :: thesis:
6577 = (37 * 177) + 28 ;
hence not 37 divides 6577 by NAT_4:9; :: thesis:
6577 = (41 * 160) + 17 ;
hence not 41 divides 6577 by NAT_4:9; :: thesis:
6577 = (43 * 152) + 41 ;
hence not 43 divides 6577 by NAT_4:9; :: thesis:
6577 = (47 * 139) + 44 ;
hence not 47 divides 6577 by NAT_4:9; :: thesis:
6577 = (53 * 124) + 5 ;
hence not 53 divides 6577 by NAT_4:9; :: thesis:
6577 = (59 * 111) + 28 ;
hence not 59 divides 6577 by NAT_4:9; :: thesis:
6577 = (61 * 107) + 50 ;
hence not 61 divides 6577 by NAT_4:9; :: thesis:
6577 = (67 * 98) + 11 ;
hence not 67 divides 6577 by NAT_4:9; :: thesis:
6577 = (71 * 92) + 45 ;
hence not 71 divides 6577 by NAT_4:9; :: thesis:
6577 = (73 * 90) + 7 ;
hence not 73 divides 6577 by NAT_4:9; :: thesis:
6577 = (79 * 83) + 20 ;
hence not 79 divides 6577 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6577 & n is prime holds
not n divides 6577 by XPRIMET1:44;
hence 6577 is prime by NAT_4:14; :: thesis: