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
  I is positive-implicative-ideal of X iff for x,y,z being Element of X
  st (x\y)\z in I & y\z in I holds x\z in I
proof
A1: 0.X in I by BCIALG_1:def 18;
  thus I is positive-implicative-ideal of X implies for x,y,z being Element of
  X st (x\y)\z in I & y\z in I holds x\z in I by Def8;
  assume
  for x,y,z being Element of X st (x\y)\z in I & y\z in I holds x\z in I;
  hence thesis by A1,Def8;
end;
