now :: thesis: ( not 2 divides 11 & not 3 divides 11 )

11 = (2 * 5) + 1 ;
hence not 2 divides 11 by NAT_4:9; :: thesis: not 3 divides 11
11 = (3 * 3) + 2 ;
hence not 3 divides 11 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 11 & n is prime holds
not n divides 11 by XPRIMET1:4;
hence 11 is prime by NAT_4:14; :: thesis: verum