reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;

theorem
  (for x being Element of X holds x\(0.X\x) in I) implies I is closed
  Ideal of X
proof
  assume
A1: for x being Element of X holds x\(0.X\x) in I;
  for x1 being Element of I holds x1` in I
  proof
    let x1 be Element of I;
    (0.X\x1)\x1=(0.X\(0.X\(0.X\x1)))\x1 by BCIALG_1:8;
    then (0.X\x1)\x1=(0.X\x1)\(0.X\(0.X\x1)) by BCIALG_1:7;
    then (0.X\x1)\x1 in I by A1;
    hence thesis by BCIALG_1:def 18;
  end;
  hence thesis by BCIALG_1:def 19;
end;
