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

theorem
  X is commutative BCK-algebra implies for x,y being Element of X holds
  (x\y=x iff y\(y\x)=0.X)
proof
  assume
A1: X is commutative BCK-algebra;
A2: for x,y being Element of X holds (y\(y\x)=0.X implies x\y=x)
  proof
    let x,y be Element of X;
    assume
A3: y\(y\x)=0.X;
    x\y = x\(x\(x\y)) by BCIALG_1:8
      .= x\0.X by A1,A3,Def1
      .= x by BCIALG_1:2;
    hence thesis;
  end;
  for x,y being Element of X holds (x\y=x implies y\(y\x)=0.X)
  proof
    let x,y be Element of X;
    assume x\y=x;
    then y\(y\x) = x\x by A1,Def1
      .= 0.X by BCIALG_1:def 5;
    hence thesis;
  end;
  hence thesis by A2;
end;
