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

theorem Th37:
  for X being bounded BCK-algebra,a being Element of X st a is
  being_greatest holds (X is BCK-implicative iff X is involutory & X is
  BCK-positive-implicative )
proof
  let X be bounded BCK-algebra;
  let a be Element of X;
  assume
A1: a is being_greatest;
  thus X is BCK-implicative implies X is involutory & X is
  BCK-positive-implicative
  by Lm1,Th34;
  assume that
A2: X is involutory and
A3: X is BCK-positive-implicative;
  for x,y being Element of X holds x\(y\x)=x
  proof
    let x,y be Element of X;
    y\a = 0.X by A1;
    then y<=a;
    then
A4: y\x <= a\x by BCIALG_1:5;
    x\(a\x) = (a\(a\x))\(a\x) by A1,A2
      .= a\(a\x) by A3,Th28
      .= x by A1,A2;
    then x <= x\(y\x) by A4,BCIALG_1:5;
    then
A5: x\(x\(y\x)) = 0.X;
    (x\(y\x))\x = (x\x)\(y\x) by BCIALG_1:7
      .= (y\x)` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
    hence thesis by A5,BCIALG_1:def 7;
  end;
  hence thesis;
end;
