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 Th45:
  for X being BCK-Algebra_with_Condition(S) holds ( X is
positive-implicative iff for x,y being Element of X holds (x <= y implies x*y =
  y) )
proof
  let X be BCK-Algebra_with_Condition(S);
A1: X is positive-implicative implies for x,y being Element of X holds (x <=
  y implies x*y = y)
  proof
    assume
A2: X is positive-implicative;
    let x,y be Element of X;
A3: (y*x)\y <= x by Lm2;
    (x*y)\y = (y*x)\y by Th6
      .= ((y*x)\y)\y by A2;
    then (x*y)\y <= x\y by A3,BCIALG_1:5;
    then
A4: ((x*y)\y) \ (x\y) = 0.X;
A5: y\(x*y) = y\(y*x) by Th6
      .= (y\y)\x by Th11
      .= x` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
    assume x <= y;
    then (x\y) = 0.X;
    then (x*y)\y = 0.X by A4,BCIALG_1:2;
    hence thesis by A5,BCIALG_1:def 7;
  end;
  (for x,y being Element of X holds (x <= y implies x*y = y)) implies X
  is positive-implicative
  proof
    assume
A6: for x,y being Element of X holds (x <= y implies x*y = y);
    for x being Element of X holds x*x = x
    proof
      let x be Element of X;
      x\x = 0.X by BCIALG_1:def 5;
      then x <= x;
      hence thesis by A6;
    end;
    hence thesis by Th44;
  end;
  hence thesis by A1;
end;
