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 Th36:
  {0.X} is commutative Ideal of X iff X is commutative BCK-algebra
proof
A1: X is commutative BCK-algebra implies for x,y being Element of X holds x\y
  = x\(y\(y\x))
  proof
    assume
A2: X is commutative BCK-algebra;
    let x,y be Element of X;
    x\y = x\(x\(x\y)) by BCIALG_1:8
      .= x\(y\(y\x)) by A2,BCIALG_3:def 1;
    hence thesis;
  end;
  thus {0.X} is commutative Ideal of X implies X is commutative BCK-algebra
  proof
    assume
A3: {0.X} is commutative Ideal of X;
    for x,y being Element of X st x<=y holds x=y\(y\x)
    proof
      let x,y be Element of X;
      assume x<=y;
      then x\y=0.X;
      then x\y in {0.X} by TARSKI:def 1;
      then x\(y\(y\x)) in {0.X} by A3,Th33;
      then (y\(y\x))\x=0.X & x\(y\(y\x))=0.X by BCIALG_1:1,TARSKI:def 1;
      hence thesis by BCIALG_1:def 7;
    end;
    hence thesis by BCIALG_3:5;
  end;
  assume X is commutative BCK-algebra;
  then for x,y being Element of X st x\y in {0.X} holds x\(y\(y\x)) in {0.X}
  by A1;
  hence thesis by Th33,BCIALG_1:43;
end;
