now :: thesis: ( not 2 divides 683 & not 3 divides 683 & not 5 divides 683 & not 7 divides 683 & not 11 divides 683 & not 13 divides 683 & not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )

683 = (2 * 341) + 1 ;
hence not 2 divides 683 by NAT_4:9; :: thesis: ( not 3 divides 683 & not 5 divides 683 & not 7 divides 683 & not 11 divides 683 & not 13 divides 683 & not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )
683 = (3 * 227) + 2 ;
hence not 3 divides 683 by NAT_4:9; :: thesis: ( not 5 divides 683 & not 7 divides 683 & not 11 divides 683 & not 13 divides 683 & not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )
683 = (5 * 136) + 3 ;
hence not 5 divides 683 by NAT_4:9; :: thesis: ( not 7 divides 683 & not 11 divides 683 & not 13 divides 683 & not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )
683 = (7 * 97) + 4 ;
hence not 7 divides 683 by NAT_4:9; :: thesis: ( not 11 divides 683 & not 13 divides 683 & not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )
683 = (11 * 62) + 1 ;
hence not 11 divides 683 by NAT_4:9; :: thesis: ( not 13 divides 683 & not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )
683 = (13 * 52) + 7 ;
hence not 13 divides 683 by NAT_4:9; :: thesis: ( not 17 divides 683 & not 19 divides 683 & not 23 divides 683 )
683 = (17 * 40) + 3 ;
hence not 17 divides 683 by NAT_4:9; :: thesis: ( not 19 divides 683 & not 23 divides 683 )
683 = (19 * 35) + 18 ;
hence not 19 divides 683 by NAT_4:9; :: thesis: not 23 divides 683
683 = (23 * 29) + 16 ;
hence not 23 divides 683 by NAT_4:9; :: thesis: verum

end;

then for n being Element of NAT st 1 < n & n * n <= 683 & n is prime holds
not n divides 683 by XPRIMET1:18;
hence 683 is prime by NAT_4:14; :: thesis: verum