now :: thesis: ( not 2 divides 223 & not 3 divides 223 & not 5 divides 223 & not 7 divides 223 & not 11 divides 223 & not 13 divides 223 )

223 = (2 * 111) + 1 ;
hence not 2 divides 223 by NAT_4:9; :: thesis: ( not 3 divides 223 & not 5 divides 223 & not 7 divides 223 & not 11 divides 223 & not 13 divides 223 )
223 = (3 * 74) + 1 ;
hence not 3 divides 223 by NAT_4:9; :: thesis: ( not 5 divides 223 & not 7 divides 223 & not 11 divides 223 & not 13 divides 223 )
223 = (5 * 44) + 3 ;
hence not 5 divides 223 by NAT_4:9; :: thesis: ( not 7 divides 223 & not 11 divides 223 & not 13 divides 223 )
223 = (7 * 31) + 6 ;
hence not 7 divides 223 by NAT_4:9; :: thesis: ( not 11 divides 223 & not 13 divides 223 )
223 = (11 * 20) + 3 ;
hence not 11 divides 223 by NAT_4:9; :: thesis: not 13 divides 223
223 = (13 * 17) + 2 ;
hence not 13 divides 223 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 223 & n is prime holds
not n divides 223 by XPRIMET1:12;
hence 223 is prime by NAT_4:14; :: thesis: verum