reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of  the carrier of X;
reserve j,i,k,n,m for Nat;
reserve R for Equivalence_Relation of X;
reserve RI for I-congruence of X,I;
reserve E for Congruence of X;
reserve RC for R-congruence of X;
reserve LC for L-congruence of X;
reserve E for Congruence of X;
reserve RI for I-congruence of X,I;
reserve W1,W2 for Element of Class E;

theorem
  for X,I st I = BCK-part(X) holds for RI being I-congruence of X,I
  holds X./.RI is p-Semisimple BCI-algebra
proof
  let X,I;
  assume
A1: I = BCK-part(X);
  let RI be I-congruence of X,I;
  set IT = X./.RI;
  for w1 being Element of IT holds (w1`)` = w1
  proof
    let w1 be Element of IT;
    w1 in the carrier of IT;
    then consider x1 being object such that
A2: x1 in the carrier of X and
A3: w1 = Class(RI,x1) by EQREL_1:def 3;
    reconsider x1 as Element of X by A2;
    w1`= Class(RI,x1`) by A3,Def17;
    then
A4: w1``=Class(RI,x1``) by Def17;
    x1\((x1`)`) is positive Element of X by Th28;
    then 0.X <= x1\(x1``) by Def2;
    then
A5: x1\(x1``) in I by A1;
    0.X in I by BCIALG_1:def 18;
    then (x1`)`\x1 in I by BCIALG_1:1;
    then [x1``,x1] in RI by A5,Def12;
    hence thesis by A3,A4,EQREL_1:35;
  end;
  hence thesis by BCIALG_1:54;
end;
