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 associative BCI-algebra,A being Ideal of X holds A is closed
proof
  let X be associative BCI-algebra;
  let A be Ideal of X;
  for x being Element of A holds x` in A
  proof
    let x be Element of A;
    0.X\x=x\0.X by BCIALG_1:48
      .=x by BCIALG_1:2;
    hence thesis;
  end;
  hence thesis;
end;
