reserve i,j,k,n for Nat;

theorem
  for f being Function holds f is one-to-one iff for x being set st x in
  dom f holds f"{f.x} = {x}
proof
  let f be Function;
  now
    hereby
      assume
A1:   for x being object st x in dom f holds f is_one-to-one_at x;
      let x be object;
      assume x in dom f;
      then f is_one-to-one_at x by A1;
      hence f"{f.x} = {x} by FINSEQ_4:2;
    end;
    assume
A2: for x being object st x in dom f holds f"{f.x} = {x};
    let x be object;
    assume
A3: x in dom f;
    then f"{f.x} = {x} by A2;
    hence f is_one-to-one_at x by A3,FINSEQ_4:2;
  end;
  hence thesis by FINSEQ_4:4;
end;
