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 BCK-algebra st X is p-Semisimple BCI-algebra holds {0.X}
  is commutative Ideal of X
proof
  let X be BCK-algebra;
  set X1={0.X};
A1: X1 c= the carrier of X
  proof
    let xx be object;
    assume xx in X1;
    then xx=0.X by TARSKI:def 1;
    hence thesis;
  end;
A2: for x,y being Element of X st x\y in {0.X} & y in {0.X} holds x in {0.X}
  proof
    set X1={0.X};
    let x,y be Element of X;
    assume x\y in X1 & y in X1;
    then x\y = 0.X & y = 0.X by TARSKI:def 1;
    then x=0.X by BCIALG_1:2;
    hence thesis by TARSKI:def 1;
  end;
  0.X in {0.X} by TARSKI:def 1;
  then reconsider X1 as Ideal of X by A1,A2,BCIALG_1:def 18;
  assume
A3: X is p-Semisimple BCI-algebra;
  for x,y,z being Element of X st (x\y)\z in X1 & z in X1 holds x\(y\(y\x
  )) in X1
  proof
    let x,y,z be Element of X;
    assume (x\y)\z in X1 & z in X1;
    then (x\y)\z=0.X & z=0.X by TARSKI:def 1;
    then
A4: x\y = 0.X by BCIALG_1:2;
    y is atom by A3,BCIALG_1:52;
    then x=y by A4;
    then x\(y\(y\x))=x\(x\(0.X)) by BCIALG_1:def 5;
    then x\(y\(y\x))=x\x by BCIALG_1:2;
    then x\(y\(y\x))=0.X by BCIALG_1:def 5;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis by Def6;
end;
