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 Th15:
  I is associative-ideal of X iff for x,y,z st (x\y)\z in I holds x\(y\z) in I
proof
  thus I is associative-ideal of X implies for x,y,z st (x\y)\z in I holds x\(
  y\z) in I
  proof
    assume
A1: I is associative-ideal of X;
    let x,y,z such that
A2: (x\y)\z in I;
    x\(x\y)\y =0.X by BCIALG_1:1;
    then x\(x\y)<= y;
    then x\(x\y)\(y\z)<= y\(y\z) by BCIALG_1:5;
    then
A3: (x\(x\y)\(y\z))\z <= (y\(y\z))\z by BCIALG_1:5;
    (y\(y\z))\z =0.X by BCIALG_1:1;
    then ((x\(x\y)\(y\z))\z)\0.X=0.X by A3;
    then ((x\(x\y))\(y\z))\z =0.X by BCIALG_1:2;
    then ((x\(x\y))\(y\z))\z in I by A1,Def4;
    then ((x\(y\z))\(x\y))\z in I by BCIALG_1:7;
    hence thesis by A1,A2,Def4;
  end;
  assume
A4: for x,y,z st (x\y)\z in I holds x\(y\z) in I;
A5: for x,y,z st (x\y)\z in I & y\z in I holds x in I
  proof
    let x,y,z such that
A6: (x\y)\z in I and
A7: y\z in I;
    x\(y\z) in I by A4,A6;
    hence thesis by A7,BCIALG_1:def 18;
  end;
  0.X in I by BCIALG_1:def 18;
  hence thesis by A5,Def4;
end;
