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
  AtomSet(X) is Ideal of X implies AtomSet(X) is closed Ideal of X
proof
  set P = AtomSet(X);
A1: for x being Element of P holds x` in P
  proof
    let x be Element of P;
    x` is minimal by BCIALG_2:30;
    hence thesis;
  end;
  assume P is Ideal of X;
  hence thesis by A1,BCIALG_1:def 19;
end;
