now :: thesis: ( not 2 divides 271 & not 3 divides 271 & not 5 divides 271 & not 7 divides 271 & not 11 divides 271 & not 13 divides 271 )

271 = (2 * 135) + 1 ;
hence not 2 divides 271 by NAT_4:9; :: thesis: ( not 3 divides 271 & not 5 divides 271 & not 7 divides 271 & not 11 divides 271 & not 13 divides 271 )
271 = (3 * 90) + 1 ;
hence not 3 divides 271 by NAT_4:9; :: thesis: ( not 5 divides 271 & not 7 divides 271 & not 11 divides 271 & not 13 divides 271 )
271 = (5 * 54) + 1 ;
hence not 5 divides 271 by NAT_4:9; :: thesis: ( not 7 divides 271 & not 11 divides 271 & not 13 divides 271 )
271 = (7 * 38) + 5 ;
hence not 7 divides 271 by NAT_4:9; :: thesis: ( not 11 divides 271 & not 13 divides 271 )
271 = (11 * 24) + 7 ;
hence not 11 divides 271 by NAT_4:9; :: thesis: not 13 divides 271
271 = (13 * 20) + 11 ;
hence not 13 divides 271 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 271 & n is prime holds
not n divides 271 by XPRIMET1:12;
hence 271 is prime by NAT_4:14; :: thesis: verum