now :: thesis: ( not 2 divides 353 & not 3 divides 353 & not 5 divides 353 & not 7 divides 353 & not 11 divides 353 & not 13 divides 353 & not 17 divides 353 )

353 = (2 * 176) + 1 ;
hence not 2 divides 353 by NAT_4:9; :: thesis: ( not 3 divides 353 & not 5 divides 353 & not 7 divides 353 & not 11 divides 353 & not 13 divides 353 & not 17 divides 353 )
353 = (3 * 117) + 2 ;
hence not 3 divides 353 by NAT_4:9; :: thesis: ( not 5 divides 353 & not 7 divides 353 & not 11 divides 353 & not 13 divides 353 & not 17 divides 353 )
353 = (5 * 70) + 3 ;
hence not 5 divides 353 by NAT_4:9; :: thesis: ( not 7 divides 353 & not 11 divides 353 & not 13 divides 353 & not 17 divides 353 )
353 = (7 * 50) + 3 ;
hence not 7 divides 353 by NAT_4:9; :: thesis: ( not 11 divides 353 & not 13 divides 353 & not 17 divides 353 )
353 = (11 * 32) + 1 ;
hence not 11 divides 353 by NAT_4:9; :: thesis: ( not 13 divides 353 & not 17 divides 353 )
353 = (13 * 27) + 2 ;
hence not 13 divides 353 by NAT_4:9; :: thesis: not 17 divides 353
353 = (17 * 20) + 13 ;
hence not 17 divides 353 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 353 & n is prime holds
not n divides 353 by XPRIMET1:14;
hence 353 is prime by NAT_4:14; :: thesis: verum