now :: thesis: ( not 2 divides 313 & not 3 divides 313 & not 5 divides 313 & not 7 divides 313 & not 11 divides 313 & not 13 divides 313 & not 17 divides 313 )

313 = (2 * 156) + 1 ;
hence not 2 divides 313 by NAT_4:9; :: thesis: ( not 3 divides 313 & not 5 divides 313 & not 7 divides 313 & not 11 divides 313 & not 13 divides 313 & not 17 divides 313 )
313 = (3 * 104) + 1 ;
hence not 3 divides 313 by NAT_4:9; :: thesis: ( not 5 divides 313 & not 7 divides 313 & not 11 divides 313 & not 13 divides 313 & not 17 divides 313 )
313 = (5 * 62) + 3 ;
hence not 5 divides 313 by NAT_4:9; :: thesis: ( not 7 divides 313 & not 11 divides 313 & not 13 divides 313 & not 17 divides 313 )
313 = (7 * 44) + 5 ;
hence not 7 divides 313 by NAT_4:9; :: thesis: ( not 11 divides 313 & not 13 divides 313 & not 17 divides 313 )
313 = (11 * 28) + 5 ;
hence not 11 divides 313 by NAT_4:9; :: thesis: ( not 13 divides 313 & not 17 divides 313 )
313 = (13 * 24) + 1 ;
hence not 13 divides 313 by NAT_4:9; :: thesis: not 17 divides 313
313 = (17 * 18) + 7 ;
hence not 17 divides 313 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 313 & n is prime holds
not n divides 313 by XPRIMET1:14;
hence 313 is prime by NAT_4:14; :: thesis: verum