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 Th12:
  ((0.X,x) to_power n)``=(0.X,x) to_power n
proof
  defpred P[set] means for m being Nat 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 being Nat st m=k & m<=n holds (((0.X,x)to_power m
    )`)`=(0.X,x)to_power m;
    let m be Nat;
    assume m=k+1 & m<=n;
    then
A2: k<=n by NAT_1:13;
    (((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
      .=(((0.X,x)to_power k)`)`\((x`)`) by BCIALG_1:9
      .=(0.X,x)to_power k\((x`)`) by A1,A2
      .=(((x`)`)`,x)to_power k by Th7
      .=(x`,x)to_power k by BCIALG_1:8
      .=(0.X,x)to_power k\x by Th7;
    hence (((0.X,x)to_power (k+1))`)` =(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
    .=(0.X)` by BCIALG_1:def 5
    .=0.X by BCIALG_1:def 5;
  then
A4: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis;
end;
