
theorem
for R,S being Ring
holds R == S iff ex f being Function of R,S st f = id R & f is isomorphism
proof
let R,S being Ring;
A: now assume B0: R == S;
   thus ex f being Function of R,S st f = id R & f is isomorphism
     proof
     reconsider f = id R as Function of R,S by B0;
     take f;
     f is additive multiplicative unity-preserving by B0;
     hence thesis by B0;
     end;
   end;
now assume ex f being Function of R,S st f = id R & f is isomorphism;
  then consider f being Function of R,S such that
  B0: f = id R & f is isomorphism;
  B1: now let o be object;
      assume o in the carrier of R;
      then reconsider a = o as Element of R;
      f.a in the carrier of S;
      hence o in the carrier of S by B0;
      end;
  BB: now let o be object;
      assume B2: o in the carrier of S;
      rng f = the carrier of S by B0,FUNCT_2:def 3;
      hence o in the carrier of R by B2,B0;
      end; then
  B2: the carrier of S = the carrier of R by B1,TARSKI:2;
  H: f is additive multiplicative unity-preserving by B0;
  B4: 0.S = f.(0.R) by B0,RING_2:6 .= 0.R by B0;
  now let x be object;
    assume x in [:the carrier of R,the carrier of R:]; then
    consider a,b being object such that
    B5: a in the carrier of R & b in the carrier of R & x = [a,b]
        by ZFMISC_1:def 2;
    reconsider a,b as Element of R by B5;
    thus (the addF of R).x = f.(a + b) by B5,B0
                          .= f.a + f.b by H
                          .= (the addF of S).x by B0,B5;
    end; then
  B5: the addF of R = the addF of S by B2;
  now let x be object;
    assume x in [:the carrier of R,the carrier of R:]; then
    consider a,b being object such that
    B5: a in the carrier of R & b in the carrier of R & x = [a,b]
        by ZFMISC_1:def 2;
    reconsider a,b as Element of R by B5;
    thus (the multF of R).x = f.(a * b) by B5,B0
                           .= f.a * f.b by H
                           .= (the multF of S).x by B0,B5;
    end; then
  the multF of R = the multF of S by B2;
  hence R == S by B1,BB,TARSKI:2,H,B0,B4,B5;
  end;
hence thesis by A;
end;
