now :: thesis: ( not 2 divides 239 & not 3 divides 239 & not 5 divides 239 & not 7 divides 239 & not 11 divides 239 & not 13 divides 239 )

239 = (2 * 119) + 1 ;
hence not 2 divides 239 by NAT_4:9; :: thesis: ( not 3 divides 239 & not 5 divides 239 & not 7 divides 239 & not 11 divides 239 & not 13 divides 239 )
239 = (3 * 79) + 2 ;
hence not 3 divides 239 by NAT_4:9; :: thesis: ( not 5 divides 239 & not 7 divides 239 & not 11 divides 239 & not 13 divides 239 )
239 = (5 * 47) + 4 ;
hence not 5 divides 239 by NAT_4:9; :: thesis: ( not 7 divides 239 & not 11 divides 239 & not 13 divides 239 )
239 = (7 * 34) + 1 ;
hence not 7 divides 239 by NAT_4:9; :: thesis: ( not 11 divides 239 & not 13 divides 239 )
239 = (11 * 21) + 8 ;
hence not 11 divides 239 by NAT_4:9; :: thesis: not 13 divides 239
239 = (13 * 18) + 5 ;
hence not 13 divides 239 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 239 & n is prime holds
not n divides 239 by XPRIMET1:12;
hence 239 is prime by NAT_4:14; :: thesis: verum