
theorem
  for R being left_unital right_unital non empty doubleLoopStr, I
  being right-ideal non empty Subset of R holds I is proper iff for u being
  Element of R st u is unital holds not(u in I)
proof
  let R be left_unital right_unital non empty doubleLoopStr, I be
  right-ideal non empty Subset of R;
A1: now
    assume
A2: I is proper;
A3: not 1.R in I
    proof
      assume
A4:   1.R in I;
A5:   now
        let u be object;
        assume u in the carrier of R;
        then reconsider u9 = u as Element of R;
        1.R*u9 = u9;
        hence u in I by A4,Def3;
      end;
      for u being object holds u in I implies u in the carrier of R;
      then I = the carrier of R by A5,TARSKI:2;
      hence thesis by A2,SUBSET_1:def 6;
    end;
    thus for u being Element of R st u is unital holds not u in I
    proof
      let u be Element of R;
      assume u is unital;
      then ex b being Element of R st 1.R = u*b & 1.R = b*u by GCD_1:def 2;
      hence thesis by A3,Def3;
    end;
  end;
  now
    1.R = 1.R * 1.R;
    then
A6: 1.R is unital by GCD_1:def 2;
    assume for u being Element of R st u is unital holds not u in I;
    then I <> the carrier of R by A6;
    hence I is proper by SUBSET_1:def 6;
  end;
  hence thesis by A1;
end;
