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 (0.X)|^n = 0.X
proof
  let X be BCI-Algebra_with_Condition(S);
  defpred P[set] means for m holds m=$1 & m<= n implies (0.X)|^m = 0.X;
A1: for k st P[k] holds P[k+1] by Lm6;
A2: P[0] by Def6;
  for n holds P[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
