6709 = (2 * 3354) + 1 ;
hence not 2 divides 6709 by NAT_4:9; :: thesis:
6709 = (3 * 2236) + 1 ;
hence not 3 divides 6709 by NAT_4:9; :: thesis:
6709 = (5 * 1341) + 4 ;
hence not 5 divides 6709 by NAT_4:9; :: thesis:
6709 = (7 * 958) + 3 ;
hence not 7 divides 6709 by NAT_4:9; :: thesis:
6709 = (11 * 609) + 10 ;
hence not 11 divides 6709 by NAT_4:9; :: thesis:
6709 = (13 * 516) + 1 ;
hence not 13 divides 6709 by NAT_4:9; :: thesis:
6709 = (17 * 394) + 11 ;
hence not 17 divides 6709 by NAT_4:9; :: thesis:
6709 = (19 * 353) + 2 ;
hence not 19 divides 6709 by NAT_4:9; :: thesis:
6709 = (23 * 291) + 16 ;
hence not 23 divides 6709 by NAT_4:9; :: thesis:
6709 = (29 * 231) + 10 ;
hence not 29 divides 6709 by NAT_4:9; :: thesis:
6709 = (31 * 216) + 13 ;
hence not 31 divides 6709 by NAT_4:9; :: thesis:
6709 = (37 * 181) + 12 ;
hence not 37 divides 6709 by NAT_4:9; :: thesis:
6709 = (41 * 163) + 26 ;
hence not 41 divides 6709 by NAT_4:9; :: thesis:
6709 = (43 * 156) + 1 ;
hence not 43 divides 6709 by NAT_4:9; :: thesis:
6709 = (47 * 142) + 35 ;
hence not 47 divides 6709 by NAT_4:9; :: thesis:
6709 = (53 * 126) + 31 ;
hence not 53 divides 6709 by NAT_4:9; :: thesis:
6709 = (59 * 113) + 42 ;
hence not 59 divides 6709 by NAT_4:9; :: thesis:
6709 = (61 * 109) + 60 ;
hence not 61 divides 6709 by NAT_4:9; :: thesis:
6709 = (67 * 100) + 9 ;
hence not 67 divides 6709 by NAT_4:9; :: thesis:
6709 = (71 * 94) + 35 ;
hence not 71 divides 6709 by NAT_4:9; :: thesis:
6709 = (73 * 91) + 66 ;
hence not 73 divides 6709 by NAT_4:9; :: thesis:
6709 = (79 * 84) + 73 ;
hence not 79 divides 6709 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 6709 & n is prime holds
not n divides 6709 by XPRIMET1:44;
hence 6709 is prime by NAT_4:14; :: thesis: