
theorem
  for R being non empty multLoopStr, I being right-ideal non empty
  Subset of R, J being Subset of R holds I c= I % J
proof
  let R be non empty multLoopStr, I be right-ideal non empty Subset of R,
  J be Subset of R;
    let u be object;
    assume
A1: u in I;
    then reconsider u9 = u as Element of R;
    now
      let v be object;
      assume v in u9*J;
      then ex j being Element of R st v = u9*j & j in J;
      hence v in I by A1,Def3;
    end;
    then u9*J c= I;
    hence u in (I % J);
end;
