reserve X for BCK-algebra;
reserve x,y for Element of X;
reserve IT for non empty Subset of X;

theorem
  IT is Commutative-Ideal of X implies for x,y being Element of X st x\(
  x\y) in IT holds (y\(y\x))\(x\y) in IT
proof
  assume IT is Commutative-Ideal of X;
  then
A1: IT is Ideal of X by Th26;
  let x,y be Element of X;
  ((y\(y\x))\(x\y))\(x\(x\y)) = ((y\(x\y))\(y\x))\(x\(x\y)) by BCIALG_1:7
    .= 0.X by BCIALG_1:11;
  then
A2: (y\(y\x))\(x\y) <= x\(x\y);
  assume x\(x\y) in IT;
  hence thesis by A1,A2,BCIALG_1:46;
end;
