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 holds BCK-part(X) is commutative Ideal of X
proof
  let X be BCK-algebra;
  set B = BCK-part(X);
A1: for x,y being Element of X st x\y in B & y in B holds x in B
  proof
    let x,y be Element of X such that
A2: x\y in B and
A3: y in B;
    ex x1 being Element of X st x\y=x1 & 0.X<=x1 by A2;
    then (x\y)`=0.X;
    then
A4: x`\y`=0.X by BCIALG_1:9;
    ex x2 being Element of X st y=x2 & 0.X<=x2 by A3;
    then x`\0.X = 0.X by A4;
    then 0.X\x=0.X by BCIALG_1:2;
    then 0.X<= x;
    hence thesis;
  end;
A5: for x,y,z being Element of X st (x\y)\z in B & z in B holds x\(y\(y\x))
  in B
  proof
    let x,y,z be Element of X;
    assume that
    (x\y)\z in B and
    z in B;
    0.X\(x\(y\(y\x)))=(x\(y\(y\x)))` .= 0.X by BCIALG_1:def 8;
    then 0.X <= (x\(y\(y\x)));
    hence thesis;
  end;
  0.X in BCK-part(X) by BCIALG_1:19;
  hence thesis by A1,A5,Def6,BCIALG_1:def 18;
end;
