now :: thesis: ( not 2 divides 331 & not 3 divides 331 & not 5 divides 331 & not 7 divides 331 & not 11 divides 331 & not 13 divides 331 & not 17 divides 331 )

331 = (2 * 165) + 1 ;
hence not 2 divides 331 by NAT_4:9; :: thesis: ( not 3 divides 331 & not 5 divides 331 & not 7 divides 331 & not 11 divides 331 & not 13 divides 331 & not 17 divides 331 )
331 = (3 * 110) + 1 ;
hence not 3 divides 331 by NAT_4:9; :: thesis: ( not 5 divides 331 & not 7 divides 331 & not 11 divides 331 & not 13 divides 331 & not 17 divides 331 )
331 = (5 * 66) + 1 ;
hence not 5 divides 331 by NAT_4:9; :: thesis: ( not 7 divides 331 & not 11 divides 331 & not 13 divides 331 & not 17 divides 331 )
331 = (7 * 47) + 2 ;
hence not 7 divides 331 by NAT_4:9; :: thesis: ( not 11 divides 331 & not 13 divides 331 & not 17 divides 331 )
331 = (11 * 30) + 1 ;
hence not 11 divides 331 by NAT_4:9; :: thesis: ( not 13 divides 331 & not 17 divides 331 )
331 = (13 * 25) + 6 ;
hence not 13 divides 331 by NAT_4:9; :: thesis: not 17 divides 331
331 = (17 * 19) + 8 ;
hence not 17 divides 331 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 331 & n is prime holds
not n divides 331 by XPRIMET1:14;
hence 331 is prime by NAT_4:14; :: thesis: verum