now :: thesis: ( not 2 divides 227 & not 3 divides 227 & not 5 divides 227 & not 7 divides 227 & not 11 divides 227 & not 13 divides 227 )

227 = (2 * 113) + 1 ;
hence not 2 divides 227 by NAT_4:9; :: thesis: ( not 3 divides 227 & not 5 divides 227 & not 7 divides 227 & not 11 divides 227 & not 13 divides 227 )
227 = (3 * 75) + 2 ;
hence not 3 divides 227 by NAT_4:9; :: thesis: ( not 5 divides 227 & not 7 divides 227 & not 11 divides 227 & not 13 divides 227 )
227 = (5 * 45) + 2 ;
hence not 5 divides 227 by NAT_4:9; :: thesis: ( not 7 divides 227 & not 11 divides 227 & not 13 divides 227 )
227 = (7 * 32) + 3 ;
hence not 7 divides 227 by NAT_4:9; :: thesis: ( not 11 divides 227 & not 13 divides 227 )
227 = (11 * 20) + 7 ;
hence not 11 divides 227 by NAT_4:9; :: thesis: not 13 divides 227
227 = (13 * 17) + 6 ;
hence not 13 divides 227 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 227 & n is prime holds
not n divides 227 by XPRIMET1:12;
hence 227 is prime by NAT_4:14; :: thesis: verum