now :: thesis: ( not 2 divides 233 & not 3 divides 233 & not 5 divides 233 & not 7 divides 233 & not 11 divides 233 & not 13 divides 233 )

233 = (2 * 116) + 1 ;
hence not 2 divides 233 by NAT_4:9; :: thesis: ( not 3 divides 233 & not 5 divides 233 & not 7 divides 233 & not 11 divides 233 & not 13 divides 233 )
233 = (3 * 77) + 2 ;
hence not 3 divides 233 by NAT_4:9; :: thesis: ( not 5 divides 233 & not 7 divides 233 & not 11 divides 233 & not 13 divides 233 )
233 = (5 * 46) + 3 ;
hence not 5 divides 233 by NAT_4:9; :: thesis: ( not 7 divides 233 & not 11 divides 233 & not 13 divides 233 )
233 = (7 * 33) + 2 ;
hence not 7 divides 233 by NAT_4:9; :: thesis: ( not 11 divides 233 & not 13 divides 233 )
233 = (11 * 21) + 2 ;
hence not 11 divides 233 by NAT_4:9; :: thesis: not 13 divides 233
233 = (13 * 17) + 12 ;
hence not 13 divides 233 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 233 & n is prime holds
not n divides 233 by XPRIMET1:12;
hence 233 is prime by NAT_4:14; :: thesis: verum