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
  X1 is associative-ideal of X implies X1 is Ideal of X
proof
  assume
A1: X1 is associative-ideal of X;
A2: for x,y being Element of X st x\y in X1&y in X1 holds x in X1
  proof
    let x,y be Element of X;
    assume x\y in X1 & y in X1;
    then (x\y)\0.X in X1 & y\0.X in X1 by BCIALG_1:2;
    hence thesis by A1,Def4;
  end;
  0.X in X1 by A1,Def4;
  hence thesis by A2,BCIALG_1:def 18;
end;
