now :: thesis: ( not 2 divides 257 & not 3 divides 257 & not 5 divides 257 & not 7 divides 257 & not 11 divides 257 & not 13 divides 257 )

257 = (2 * 128) + 1 ;
hence not 2 divides 257 by NAT_4:9; :: thesis: ( not 3 divides 257 & not 5 divides 257 & not 7 divides 257 & not 11 divides 257 & not 13 divides 257 )
257 = (3 * 85) + 2 ;
hence not 3 divides 257 by NAT_4:9; :: thesis: ( not 5 divides 257 & not 7 divides 257 & not 11 divides 257 & not 13 divides 257 )
257 = (5 * 51) + 2 ;
hence not 5 divides 257 by NAT_4:9; :: thesis: ( not 7 divides 257 & not 11 divides 257 & not 13 divides 257 )
257 = (7 * 36) + 5 ;
hence not 7 divides 257 by NAT_4:9; :: thesis: ( not 11 divides 257 & not 13 divides 257 )
257 = (11 * 23) + 4 ;
hence not 11 divides 257 by NAT_4:9; :: thesis: not 13 divides 257
257 = (13 * 19) + 10 ;
hence not 13 divides 257 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 257 & n is prime holds
not n divides 257 by XPRIMET1:12;
hence 257 is prime by NAT_4:14; :: thesis: verum