now :: thesis: ( not 2 divides 277 & not 3 divides 277 & not 5 divides 277 & not 7 divides 277 & not 11 divides 277 & not 13 divides 277 )

277 = (2 * 138) + 1 ;
hence not 2 divides 277 by NAT_4:9; :: thesis: ( not 3 divides 277 & not 5 divides 277 & not 7 divides 277 & not 11 divides 277 & not 13 divides 277 )
277 = (3 * 92) + 1 ;
hence not 3 divides 277 by NAT_4:9; :: thesis: ( not 5 divides 277 & not 7 divides 277 & not 11 divides 277 & not 13 divides 277 )
277 = (5 * 55) + 2 ;
hence not 5 divides 277 by NAT_4:9; :: thesis: ( not 7 divides 277 & not 11 divides 277 & not 13 divides 277 )
277 = (7 * 39) + 4 ;
hence not 7 divides 277 by NAT_4:9; :: thesis: ( not 11 divides 277 & not 13 divides 277 )
277 = (11 * 25) + 2 ;
hence not 11 divides 277 by NAT_4:9; :: thesis: not 13 divides 277
277 = (13 * 21) + 4 ;
hence not 13 divides 277 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 277 & n is prime holds
not n divides 277 by XPRIMET1:12;
hence 277 is prime by NAT_4:14; :: thesis: verum