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

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