reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem
for p being Prime,
    F being Field
holds Char F = p iff PrimeField F, Z/p are_isomorphic
proof
let p be Prime, F be Field;
A1: now assume Char F = p;
   then F is p-characteristic;
   hence PrimeField F,Z/p are_isomorphic by Th107;
   end;
now assume PrimeField F, Z/p are_isomorphic;
  then ex f being Function of Z/p,PrimeField F st f is RingIsomorphism
    by QUOFIELD:def 23;
  then PrimeField F is Z/p-isomorphic;
  then A2: PrimeField F is p-characteristic;
  PrimeField F is Subring of F by Lm1;
  hence Char F = p by A2,Th88;
  end;
hence thesis by A1;
end;
