now :: thesis: ( not 2 divides 269 & not 3 divides 269 & not 5 divides 269 & not 7 divides 269 & not 11 divides 269 & not 13 divides 269 )

269 = (2 * 134) + 1 ;
hence not 2 divides 269 by NAT_4:9; :: thesis: ( not 3 divides 269 & not 5 divides 269 & not 7 divides 269 & not 11 divides 269 & not 13 divides 269 )
269 = (3 * 89) + 2 ;
hence not 3 divides 269 by NAT_4:9; :: thesis: ( not 5 divides 269 & not 7 divides 269 & not 11 divides 269 & not 13 divides 269 )
269 = (5 * 53) + 4 ;
hence not 5 divides 269 by NAT_4:9; :: thesis: ( not 7 divides 269 & not 11 divides 269 & not 13 divides 269 )
269 = (7 * 38) + 3 ;
hence not 7 divides 269 by NAT_4:9; :: thesis: ( not 11 divides 269 & not 13 divides 269 )
269 = (11 * 24) + 5 ;
hence not 11 divides 269 by NAT_4:9; :: thesis: not 13 divides 269
269 = (13 * 20) + 9 ;
hence not 13 divides 269 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 269 & n is prime holds
not n divides 269 by XPRIMET1:12;
hence 269 is prime by NAT_4:14; :: thesis: verum