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) to_power n = x\(a|^n)
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,a be Element of X;
  defpred P[set] means for m holds m=$1 & m<= n implies (x,a) to_power m = x\(
  a|^m);
  now
    let k;
    assume
A1: for m st m=k & m<= n holds (x,a) to_power m = x\(a|^m);
    let m;
    assume that
A2: m=k+1 and
A3: m<=n;
A4: (x,a) to_power m = (x,a) to_power k \ a & k<=n by A2,A3,BCIALG_2:4,NAT_1:13
;
    x\(a|^m) = x\((a|^k)*a) by A2,Def6
      .= (x\(a|^k))\a by Th11;
    hence (x,a) to_power m = x\(a|^m) by A1,A4;
  end;
  then
A5: for k st P[k] holds P[k+1];
A6: P[0] by Lm7;
  for n holds P[n] from NAT_1:sch 2(A6,A5);
  hence thesis;
end;
