theorem Th5:
  rng a = {r} & rng b = {r} implies a = b
proof
  assume that
A1: rng a = {r} and
A2: rng b = {r};
  now
    let n;
    a.n in rng a by VALUED_0:28;
    then
A3: a.n = r by A1,TARSKI:def 1;
    b.n in rng b by VALUED_0:28;
    hence a.n = b.n by A2,A3,TARSKI:def 1;
  end;
  hence thesis;
end;
