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
  X is p-Semisimple implies BCK-part(X)={0.X}
proof
  assume
A1: X is p-Semisimple;
  thus BCK-part(X) c= {0.X}
  proof
    let x be object;
    assume
A2: x in BCK-part(X);
    then
A3: ex x1 being Element of X st x=x1 & 0.X<=x1;
    reconsider x as Element of X by A2;
    (x`)`=x by A1;
    then (0.X)`=x by A3;
    then x=0.X by Def5;
    hence thesis by TARSKI:def 1;
  end;
  thus {0.X} c= BCK-part(X)
  proof
    let x be object;
    assume
A4: x in {0.X};
    then reconsider x as Element of X by TARSKI:def 1;
    x=0.X by A4,TARSKI:def 1;
    then x`=0.X by Def5;
    then 0.X <= x;
    hence thesis;
  end;
end;
