1709 = (2 * 854) + 1 ;
hence not 2 divides 1709 by NAT_4:9; :: thesis:
1709 = (3 * 569) + 2 ;
hence not 3 divides 1709 by NAT_4:9; :: thesis:
1709 = (5 * 341) + 4 ;
hence not 5 divides 1709 by NAT_4:9; :: thesis:
1709 = (7 * 244) + 1 ;
hence not 7 divides 1709 by NAT_4:9; :: thesis:
1709 = (11 * 155) + 4 ;
hence not 11 divides 1709 by NAT_4:9; :: thesis:
1709 = (13 * 131) + 6 ;
hence not 13 divides 1709 by NAT_4:9; :: thesis:
1709 = (17 * 100) + 9 ;
hence not 17 divides 1709 by NAT_4:9; :: thesis:
1709 = (19 * 89) + 18 ;
hence not 19 divides 1709 by NAT_4:9; :: thesis:
1709 = (23 * 74) + 7 ;
hence not 23 divides 1709 by NAT_4:9; :: thesis:
1709 = (29 * 58) + 27 ;
hence not 29 divides 1709 by NAT_4:9; :: thesis:
1709 = (31 * 55) + 4 ;
hence not 31 divides 1709 by NAT_4:9; :: thesis:
1709 = (37 * 46) + 7 ;
hence not 37 divides 1709 by NAT_4:9; :: thesis:
1709 = (41 * 41) + 28 ;
hence not 41 divides 1709 by NAT_4:9; :: thesis:

end;

then for n being Element of NAT st 1 < n & n * n <= 1709 & n is prime holds
not n divides 1709 by XPRIMET1:26;
hence 1709 is prime by NAT_4:14; :: thesis: