theorem
for F being Field
holds Char F = 0 iff PrimeField F, F_Rat are_isomorphic
proof
let F be Field;
A1: now assume Char F = 0;
   then F is 0-characteristic;
   hence PrimeField F, F_Rat are_isomorphic by Th100;
   end;
now assume PrimeField F, F_Rat are_isomorphic;
  then ex f being Function of F_Rat,PrimeField F st f is RingIsomorphism
    by QUOFIELD:def 23;
  then PrimeField F is F_Rat-isomorphic;
  then A2: PrimeField F is 0-characteristic;
  PrimeField F is Subring of F by Lm1;
  hence Char F = 0 by A2,Th88;
  end;
hence thesis by A1;
end;
