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

theorem Th26:
  for X being BCK-algebra st IT is Commutative-Ideal of X holds IT
  is Ideal of X
proof
  let X be BCK-algebra;
  assume
A1: IT is Commutative-Ideal of X;
A2: for x,y being Element of X st x\y in IT & y in IT holds x in IT
  proof
    let x,y be Element of X;
    assume that
A3: x\y in IT and
A4: y in IT;
A5: x\(0.X\(0.X\x)) = x\(0.X\x`) .= x\(0.X\0.X) by BCIALG_1:def 8
      .= x\0.X by BCIALG_1:def 5
      .= x by BCIALG_1:2;
    (x\0.X)\y in IT by A3,BCIALG_1:2;
    hence thesis by A1,A4,A5,Def10;
  end;
  0.X in IT by A1,Def10;
  hence thesis by A1,A2,BCIALG_1:def 18;
end;
