now :: thesis: ( not 2 divides 17 & not 3 divides 17 )

17 = (2 * 8) + 1 ;
hence not 2 divides 17 by NAT_4:9; :: thesis: not 3 divides 17
17 = (3 * 5) + 2 ;
hence not 3 divides 17 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 17 & n is prime holds
not n divides 17 by XPRIMET1:4;
hence 17 is prime by NAT_4:14; :: thesis: verum