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 Th42:
  X is p-Semisimple BCI-algebra implies X is BCI-algebra of 0,1,0, 0
proof
  assume
A1: X is p-Semisimple BCI-algebra;
  for x,y being Element of X holds Polynom (0,1,x,y) = Polynom (0,0,y,x)
  proof
    let x,y be Element of X;
    ((x,(x\y)) to_power 1,(y\x)) to_power 1 = (x\(x\y),(y\x)) to_power 1
    by BCIALG_2:2
      .= (y,(y\x)) to_power 1 by A1,BCIALG_1:def 26
      .= ((y,(y\x)) to_power 1,(x\y)) to_power 0 by BCIALG_2:1;
    hence thesis;
  end;
  hence thesis by Def3;
end;
