reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;
reserve X for BCK-algebra;
reserve X for BCI-algebra;
reserve X for BCK-algebra;
reserve I for Ideal of X;
reserve I for Ideal of X;
reserve X for BCK-algebra;
reserve I for Ideal of X;

theorem Th53:
  (for x,y,z being Element of X st (x\y)\z in I holds (x\z)\(y\z)
  in I) implies I is positive-implicative-ideal of X
proof
  assume
A1: for x,y,z being Element of X st (x\y)\z in I holds (x\z)\(y\z) in I;
A2: for x,y,z being Element of X st (x\y)\z in I & y\z in I holds x\z in I
  proof
    let x,y,z be Element of X;
    assume that
A3: (x\y)\z in I and
A4: y\z in I;
    (x\z)\(y\z) in I by A1,A3;
    hence thesis by A4,BCIALG_1:def 18;
  end;
  0.X in I by BCIALG_1:def 18;
  hence thesis by A2,Def8;
end;
