now :: thesis: ( not 2 divides 337 & not 3 divides 337 & not 5 divides 337 & not 7 divides 337 & not 11 divides 337 & not 13 divides 337 & not 17 divides 337 )

337 = (2 * 168) + 1 ;
hence not 2 divides 337 by NAT_4:9; :: thesis: ( not 3 divides 337 & not 5 divides 337 & not 7 divides 337 & not 11 divides 337 & not 13 divides 337 & not 17 divides 337 )
337 = (3 * 112) + 1 ;
hence not 3 divides 337 by NAT_4:9; :: thesis: ( not 5 divides 337 & not 7 divides 337 & not 11 divides 337 & not 13 divides 337 & not 17 divides 337 )
337 = (5 * 67) + 2 ;
hence not 5 divides 337 by NAT_4:9; :: thesis: ( not 7 divides 337 & not 11 divides 337 & not 13 divides 337 & not 17 divides 337 )
337 = (7 * 48) + 1 ;
hence not 7 divides 337 by NAT_4:9; :: thesis: ( not 11 divides 337 & not 13 divides 337 & not 17 divides 337 )
337 = (11 * 30) + 7 ;
hence not 11 divides 337 by NAT_4:9; :: thesis: ( not 13 divides 337 & not 17 divides 337 )
337 = (13 * 25) + 12 ;
hence not 13 divides 337 by NAT_4:9; :: thesis: not 17 divides 337
337 = (17 * 19) + 14 ;
hence not 17 divides 337 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 337 & n is prime holds
not n divides 337 by XPRIMET1:14;
hence 337 is prime by NAT_4:14; :: thesis: verum