reserve X for BCI-algebra;
reserve x,y,z for Element of X;
reserve i,j,k,l,m,n for Nat;
reserve f,g for sequence of the carrier of X;
reserve B,P for non empty Subset of X;

theorem
  for X being BCI-algebra st B = BCK-part(X) & P = p-Semisimple-part(X)
  holds B /\ P = {0.X}
proof
  let X be BCI-algebra;
  assume that
A1: B = BCK-part(X) and
A2: P = p-Semisimple-part(X);
  thus B /\ P c= {0.X}
  proof
    let x be object;
    assume
A3: x in B /\ P;
    then
A4: x in P by XBOOLE_0:def 4;
A5: x in B by A3,XBOOLE_0:def 4;
    then
A6: ex x1 being Element of X st x=x1 & 0.X<=x1 by A1;
    reconsider x as Element of X by A1,A5;
A7: 0.X\x=0.X by A6;
    ex x2 being Element of X st x=x2 & x2 is minimal by A2,A4;
    then x=0.X by A7;
    hence thesis by TARSKI:def 1;
  end;
  0.X in BCK-part(X) & 0.X in p-Semisimple-part(X) by BCIALG_1:19;
  then 0.X in B /\ P by A1,A2,XBOOLE_0:def 4;
  then for x being object st x in {0.X} holds x in B /\ P by TARSKI:def 1;
  hence thesis;
end;
