reserve X for BCI-algebra;
reserve x,y,z,u,a,b for Element of X;
reserve IT for non empty Subset of X;

theorem
  BCK-part(X) is closed Ideal of X
proof
  set X1=BCK-part(X);
A1: for x,y being Element of X st x\y in X1 & y in X1 holds x in X1
  proof
    let x,y be Element of X such that
A2: x\y in X1 and
A3: y in X1;
    ex x1 being Element of X st x\y=x1 & 0.X<= x1 by A2;
    then (x\y)`=0.X;
    then
A4: x`\y`=0.X by Th9;
    ex x2 being Element of X st y=x2 & 0.X<= x2 by A3;
    then x`\0.X = 0.X by A4;
    then x`=0.X by Th2;
    then 0.X<= x;
    hence thesis;
  end;
  0.X in BCK-part(X) by Th19;
  then reconsider X1 as Ideal of X by A1,Def18;
  now
    let x be Element of X1;
    x in X1;
    then ex x1 being Element of X st x=x1 & 0.X<= x1;
    then (x`)=0.X;
    then (x`)`=0.X by Def5;
    then 0.X <=x`;
    hence x` in X1;
  end;
  hence thesis by Def19;
end;
