
theorem
  for R being left_unital non empty doubleLoopStr, I being Subset of R
  holds 1.R*I = I
proof
  let R be left_unital non empty doubleLoopStr, I be Subset of R;
A1: now
    let u be object;
    assume
A2: u in I;
    then reconsider u9 = u as Element of R;
    1.R*u9 = u9;
    hence u in 1.R*I by A2;
  end;
  now
    let u be object;
    assume u in 1.R*I;
    then ex i being Element of R st u = 1.R*i & i in I;
    hence u in I;
  end;
  hence thesis by A1,TARSKI:2;
end;
