reserve X for non empty BCIStr_1;
reserve d for Element of X;
reserve n,m,k for Nat;
reserve f for sequence of  the carrier of X;

theorem Th44:
  for X being BCK-Algebra_with_Condition(S) holds ( X is
  positive-implicative iff for x being Element of X holds x*x = x )
proof
  let X be BCK-Algebra_with_Condition(S);
A1: X is positive-implicative implies for x being Element of X holds x*x = x
  proof
    assume
A2: X is positive-implicative;
    let x be Element of X;
A3: (x*x)\x <= x by Lm2;
    (x*x)\x = ((x*x)\x)\x by A2;
    then (x*x)\x <= x\x by A3,BCIALG_1:5;
    then x\x = 0.X & ((x*x)\x) \ (x\x) = 0.X by BCIALG_1:def 5;
    then
A4: (x*x)\x = 0.X by BCIALG_1:2;
    x\(x*x) = (x\x)\x by Th11
      .= x` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
    hence thesis by A4,BCIALG_1:def 7;
  end;
  (for x being Element of X holds x*x = x) implies X is positive-implicative
  proof
    assume
A5: for x being Element of X holds x*x = x;
    for x,y being Element of X holds (x\y)\y = x\y
    proof
      let x,y be Element of X;
      x\(y*y) = x\y by A5;
      hence thesis by Th11;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
