now :: thesis: ( not 2 divides 263 & not 3 divides 263 & not 5 divides 263 & not 7 divides 263 & not 11 divides 263 & not 13 divides 263 )

263 = (2 * 131) + 1 ;
hence not 2 divides 263 by NAT_4:9; :: thesis: ( not 3 divides 263 & not 5 divides 263 & not 7 divides 263 & not 11 divides 263 & not 13 divides 263 )
263 = (3 * 87) + 2 ;
hence not 3 divides 263 by NAT_4:9; :: thesis: ( not 5 divides 263 & not 7 divides 263 & not 11 divides 263 & not 13 divides 263 )
263 = (5 * 52) + 3 ;
hence not 5 divides 263 by NAT_4:9; :: thesis: ( not 7 divides 263 & not 11 divides 263 & not 13 divides 263 )
263 = (7 * 37) + 4 ;
hence not 7 divides 263 by NAT_4:9; :: thesis: ( not 11 divides 263 & not 13 divides 263 )
263 = (11 * 23) + 10 ;
hence not 11 divides 263 by NAT_4:9; :: thesis: not 13 divides 263
263 = (13 * 20) + 3 ;
hence not 13 divides 263 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 263 & n is prime holds
not n divides 263 by XPRIMET1:12;
hence 263 is prime by NAT_4:14; :: thesis: verum