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
  X is associative BCI-algebra implies X is BCI-algebra of 0,1,0,0 & X
  is BCI-algebra of 1,0,0,0
proof
  assume
A1: X is associative BCI-algebra;
  for x being Element of X holds x`` = x
  proof
    let x be Element of X;
    x` = x by A1,BCIALG_1:47;
    hence thesis;
  end;
  then
A2: X is p-Semisimple by BCIALG_1:54;
  for x,y being Element of X holds Polynom (1,0,x,y) = Polynom (0,0,y,x)
  proof
    let x,y be Element of X;
    x\(x\y) = y by A2;
    then
A3: (x\(x\y))\(x\y) = y\(y\x) by A1,BCIALG_1:48;
    ((x,(x\y)) to_power (1+1),(y\x)) to_power 0 = (x,(x\y)) to_power 2 by
BCIALG_2:1
      .= (x\(x\y))\(x\y) by BCIALG_2:3
      .= (y,(y\x)) to_power 1 by A3,BCIALG_2:2
      .= ((y,(y\x)) to_power 1,(x\y)) to_power 0 by BCIALG_2:1;
    hence thesis;
  end;
  hence thesis by A2,Def3,Th42;
end;
