
theorem Th19:
  for R being right_unital non empty doubleLoopStr, I being
  left-ideal non empty Subset of R holds I is proper iff not(1.R in I)
proof
  let R be right_unital non empty doubleLoopStr, I be left-ideal non empty
  Subset of R;
A1: now
    assume
A2: I is proper;
    thus not 1.R in I
    proof
      assume
A3:   1.R in I;
A4:   now
        let u be object;
        assume u in the carrier of R;
        then reconsider u9 = u as Element of R;
        u9*1.R = u9;
        hence u in I by A3,Def2;
      end;
      for u being object holds u in I implies u in the carrier of R;
      then I = the carrier of R by A4,TARSKI:2;
      hence thesis by A2,SUBSET_1:def 6;
    end;
  end;
  now
    assume not 1.R in I;
    then I <> the carrier of R;
    hence I is proper by SUBSET_1:def 6;
  end;
  hence thesis by A1;
end;
