now :: thesis: ( not 2 divides 23 & not 3 divides 23 )

23 = (2 * 11) + 1 ;
hence not 2 divides 23 by NAT_4:9; :: thesis: not 3 divides 23
23 = (3 * 7) + 2 ;
hence not 3 divides 23 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 23 & n is prime holds
not n divides 23 by XPRIMET1:4;
hence 23 is prime by NAT_4:14; :: thesis: verum