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
  for I,A being Ideal of X st I c= A & I is positive-implicative-ideal
  of X holds A is positive-implicative-ideal of X
proof
  let I,A be Ideal of X;
  assume that
A1: I c= A and
A2: I is positive-implicative-ideal of X;
  for x,y being Element of X st (x\y)\y in A holds x\y in A
  proof
    let x,y be Element of X;
    (x\((x\y)\y)\y) \y = ((x\y)\((x\y)\y))\y by BCIALG_1:7
      .=(x\y\y)\ ((x\y)\y) by BCIALG_1:7
      .=0.X by BCIALG_1:def 5;
    then (x\((x\y)\y)\y) \y in I by BCIALG_1:def 18;
    then (x\((x\y)\y)\y) in I by A2,Th51;
    then
A3: (x\y)\((x\y)\y) in I by BCIALG_1:7;
    assume (x\y)\y in A;
    hence thesis by A1,A3,BCIALG_1:def 18;
  end;
  hence thesis by Th51;
end;
