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

theorem
  for X being commutative BCK-algebra holds X is BCI-commutative
  BCI-algebra & X is BCI-weakly-commutative BCI-algebra
proof
  let X be commutative BCK-algebra;
A1: for x,y being Element of X holds (x\(x\y))\(0.X\(x\y)) = y\(y\x)
  proof
    let x,y be Element of X;
A2: (0.X\(x\y)) = (x\y)` .= 0.X by BCIALG_1:def 8;
    (x\(x\y)) = y\(y\x) by Def1;
    hence thesis by A2,BCIALG_1:2;
  end;
  for x,y being Element of X st x\y=0.X holds x = y\(y\x)
  by BCIALG_1:def 11,Th5;
  hence thesis by A1,Def4,Def5;
end;
