now :: thesis: ( not 2 divides 349 & not 3 divides 349 & not 5 divides 349 & not 7 divides 349 & not 11 divides 349 & not 13 divides 349 & not 17 divides 349 )

349 = (2 * 174) + 1 ;
hence not 2 divides 349 by NAT_4:9; :: thesis: ( not 3 divides 349 & not 5 divides 349 & not 7 divides 349 & not 11 divides 349 & not 13 divides 349 & not 17 divides 349 )
349 = (3 * 116) + 1 ;
hence not 3 divides 349 by NAT_4:9; :: thesis: ( not 5 divides 349 & not 7 divides 349 & not 11 divides 349 & not 13 divides 349 & not 17 divides 349 )
349 = (5 * 69) + 4 ;
hence not 5 divides 349 by NAT_4:9; :: thesis: ( not 7 divides 349 & not 11 divides 349 & not 13 divides 349 & not 17 divides 349 )
349 = (7 * 49) + 6 ;
hence not 7 divides 349 by NAT_4:9; :: thesis: ( not 11 divides 349 & not 13 divides 349 & not 17 divides 349 )
349 = (11 * 31) + 8 ;
hence not 11 divides 349 by NAT_4:9; :: thesis: ( not 13 divides 349 & not 17 divides 349 )
349 = (13 * 26) + 11 ;
hence not 13 divides 349 by NAT_4:9; :: thesis: not 17 divides 349
349 = (17 * 20) + 9 ;
hence not 17 divides 349 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 349 & n is prime holds
not n divides 349 by XPRIMET1:14;
hence 349 is prime by NAT_4:14; :: thesis: verum