now :: thesis: ( not 2 divides 13 & not 3 divides 13 )

13 = (2 * 6) + 1 ;
hence not 2 divides 13 by NAT_4:9; :: thesis: not 3 divides 13
13 = (3 * 4) + 1 ;
hence not 3 divides 13 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 13 & n is prime holds
not n divides 13 by XPRIMET1:4;
hence 13 is prime by NAT_4:14; :: thesis: verum