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) holds (X is p-Semisimple
  BCI-algebra iff B = {0.X})
proof
  let X be BCI-algebra;
  assume
A1: B = BCK-part(X);
  thus X is p-Semisimple BCI-algebra implies B = {0.X}
  proof
    assume
A2: X is p-Semisimple BCI-algebra;
    thus B c= {0.X}
    proof
      let x be object;
      assume
A3:   x in B;
      then
A4:   ex x1 being Element of X st x=x1 & 0.X<=x1 by A1;
      reconsider x as Element of X by A1,A3;
      (x`)`=x by A2,BCIALG_1:def 26;
      then (0.X)`=x by A4;
      then x=0.X by BCIALG_1:def 5;
      hence thesis by TARSKI:def 1;
    end;
    let x be object;
    assume
A5: x in {0.X};
    then reconsider x as Element of X by TARSKI:def 1;
    x=0.X by A5,TARSKI:def 1;
    then x`=0.X by BCIALG_1:def 5;
    then 0.X <= x;
    hence thesis by A1;
  end;
  assume
A6: B = {0.X};
  for x being Element of X holds x`` = x
  proof
    let x be Element of X;
    0.X\(x\(0.X\(0.X\x))) = (0.X,(x\(0.X\(0.X\x)))) to_power 1 by BCIALG_2:2
      .= ((0.X,x) to_power 1)\((0.X,(0.X\(0.X\x))) to_power 1) by BCIALG_2:18
      .= ((0.X,x) to_power 1)\((0.X,x) to_power 1) by BCIALG_2:8
      .= 0.X by BCIALG_1:def 5;
    then 0.X <= (x\(0.X\(0.X\x)));
    then (x\(0.X\(0.X\x))) in B by A1;
    then
A7: x\(0.X\(0.X\x)) = 0.X by A6,TARSKI:def 1;
    (0.X\(0.X\x))\x = (0.X\x)\(0.X\x) by BCIALG_1:7
      .= 0.X by BCIALG_1:def 5;
    hence thesis by A7,BCIALG_1:def 7;
  end;
  hence thesis by BCIALG_1:54;
end;
