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

theorem Th23:
  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\(a\y) = 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\(a\y) = y\(a\x)
  proof
    assume
A1: X is involutory;
    let a be Element of X;
    assume
A2: a is being_greatest;
    let x,y be Element of X;
    x\(a\y) = (a\(a\y))\(a\x) & y\(a\x) = (a\(a\x))\(a\y) by A1,A2,Th22;
    hence thesis by BCIALG_1:7;
  end;
  assume
A3: for a being Element of X st a is being_greatest holds for x,y being
  Element of X holds x\(a\y) = y\(a\x);
  let a be Element of X;
  assume
A4: a is being_greatest;
  let x be Element of X;
  a\(a\x) = x\(a\a) by A3,A4
    .= x\0.X by BCIALG_1:def 5
    .= x by BCIALG_1:2;
  hence thesis;
end;
