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;
reserve X for BCK-algebra;
reserve X for BCI-algebra;
reserve X for BCK-algebra;
reserve I for Ideal of X;

theorem Th33:
  I is commutative Ideal of X iff for x,y being Element of X st x\
  y in I holds x\(y\(y\x)) in I
proof
  thus I is commutative Ideal of X implies for x,y being Element of X st x\y
  in I holds x\(y\(y\x)) in I
  proof
A1: 0.X in I by BCIALG_1:def 18;
    assume
A2: I is commutative Ideal of X;
    let x,y be Element of X;
    assume x\y in I;
    then (x\y)\0.X in I by BCIALG_1:2;
    hence thesis by A2,A1,Def6;
  end;
  assume
A3: for x,y being Element of X st x\y in I holds x\(y\(y\x)) in I;
  for x,y,z being Element of X st (x\y)\z in I & z in I holds x\(y\(y\x)) in I
  proof
    let x,y,z be Element of X;
    assume (x\y)\z in I & z in I;
    then x\y in I by BCIALG_1:def 18;
    hence thesis by A3;
  end;
  hence thesis by Def6;
end;
