now :: thesis: ( not 2 divides 311 & not 3 divides 311 & not 5 divides 311 & not 7 divides 311 & not 11 divides 311 & not 13 divides 311 & not 17 divides 311 )

311 = (2 * 155) + 1 ;
hence not 2 divides 311 by NAT_4:9; :: thesis: ( not 3 divides 311 & not 5 divides 311 & not 7 divides 311 & not 11 divides 311 & not 13 divides 311 & not 17 divides 311 )
311 = (3 * 103) + 2 ;
hence not 3 divides 311 by NAT_4:9; :: thesis: ( not 5 divides 311 & not 7 divides 311 & not 11 divides 311 & not 13 divides 311 & not 17 divides 311 )
311 = (5 * 62) + 1 ;
hence not 5 divides 311 by NAT_4:9; :: thesis: ( not 7 divides 311 & not 11 divides 311 & not 13 divides 311 & not 17 divides 311 )
311 = (7 * 44) + 3 ;
hence not 7 divides 311 by NAT_4:9; :: thesis: ( not 11 divides 311 & not 13 divides 311 & not 17 divides 311 )
311 = (11 * 28) + 3 ;
hence not 11 divides 311 by NAT_4:9; :: thesis: ( not 13 divides 311 & not 17 divides 311 )
311 = (13 * 23) + 12 ;
hence not 13 divides 311 by NAT_4:9; :: thesis: not 17 divides 311
311 = (17 * 18) + 5 ;
hence not 17 divides 311 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 311 & n is prime holds
not n divides 311 by XPRIMET1:14;
hence 311 is prime by NAT_4:14; :: thesis: verum