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;

theorem
  I is p-ideal of X iff for x,y,z st (x\z)\(y\z) in I holds x\y in I
proof
  thus I is p-ideal of X implies for x,y,z st (x\z)\(y\z) in I holds x\y in I
  proof
    assume
A1: I is p-ideal of X;
    let x,y,z such that
A2: (x\z)\(y\z) in I;
    (x\z)\(y\z)\(x\y) = 0.X by BCIALG_1:def 3;
    then (x\z)\(y\z) <= (x\y);
    hence thesis by A1,A2,Th21;
  end;
  assume
A3: for x,y,z st (x\z)\(y\z) in I holds x\y in I;
A4: for x,y,z st (x\z)\(y\z) in I & y in I holds x in I
  proof
    let x,y,z such that
A5: (x\z)\(y\z) in I and
A6: y in I;
    x\y in I by A3,A5;
    hence thesis by A6,BCIALG_1:def 18;
  end;
  0.X in I by BCIALG_1:def 18;
  hence thesis by A4,Def5;
end;
