now :: thesis: ( not 2 divides 19 & not 3 divides 19 )

19 = (2 * 9) + 1 ;
hence not 2 divides 19 by NAT_4:9; :: thesis: not 3 divides 19
19 = (3 * 6) + 1 ;
hence not 3 divides 19 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 19 & n is prime holds
not n divides 19 by XPRIMET1:4;
hence 19 is prime by NAT_4:14; :: thesis: verum