reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of  the carrier of X;
reserve j,i,k,n,m for Nat;

theorem Th9:
  ((0.X,x) to_power n)` = (0.X,x`) to_power n
proof
  defpred P[set] means for m holds m=$1 & m<= n implies ((0.X,x) to_power m)`
  = (0.X,x`) to_power m;
  now
    let k;
    assume
A1: for m holds m=k & m<= n implies ((0.X,x) to_power m)` = (0.X,x`)
    to_power m;
    let m;
A2: ((0.X,x) to_power (k+1))`=((0.X,x) to_power k\x)` by Th4
      .=((0.X,x) to_power k)`\x` by BCIALG_1:9;
    assume m=k+1 & m<=n;
    then k<=n by NAT_1:13;
    hence ((0.X,x) to_power (k+1))` = ((0.X,x`) to_power k)\x` by A1,A2
      .=(0.X,x`) to_power (k+1) by Th4;
  end;
  then
A3: for k st P[k] holds P[k+1];
  ((0.X,x) to_power 0)`= (0.X)` by Th1;
  then ((0.X,x) to_power 0)`= 0.X by BCIALG_1:2;
  then
A4: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis;
end;
