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

theorem Th22:
  for X being bounded BCK-algebra holds (X is involutory iff for a
  being Element of X st a is being_greatest holds for x,y being Element of X
  holds x\y = (a\y)\(a\x) )
proof
  let X be bounded BCK-algebra;
  thus X is involutory implies for a being Element of X st a is being_greatest
  holds for x,y being Element of X holds x\y = (a\y)\(a\x)
  proof
    assume
A1: X is involutory;
    let a be Element of X;
    assume
A2: a is being_greatest;
    for x,y being Element of X holds x\y = (a\y)\(a\x)
    proof
      let x,y be Element of X;
      x\y = (a\(a\x))\y by A1,A2
        .= (a\y)\(a\x) by BCIALG_1:7;
      hence thesis;
    end;
    hence thesis;
  end;
  assume
A3: for a being Element of X st a is being_greatest holds for x,y being
  Element of X holds x\y = (a\y)\(a\x);
  let a be Element of X;
  assume
A4: a is being_greatest;
  now
    let x be Element of X;
    (a\(a\x))\0.X = (a\0.X)\(a\x) by BCIALG_1:7
      .= x\0.X by A3,A4
      .= x by BCIALG_1:2;
    hence a\(a\x)=x by BCIALG_1:2;
  end;
  hence thesis;
end;
