reserve X for non empty BCIStr_1;
reserve d for Element of X;
reserve n,m,k for Nat;
reserve f for sequence of  the carrier of X;

theorem
  for X being BCI-Algebra_with_Condition(S) holds for x,a being Element
  of X holds ((x\a)\a)\a = x\(a|^3)
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,a be Element of X;
  (x\a)\a = x\(a*a) by Th11;
  then ((x\a)\a)\a = x\((a*a)*a) by Th11
    .= x\((a|^2)*a) by Th22
    .= x\(a|^(2+1)) by Def6;
  hence thesis;
end;
