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
  (x,x\(x\y)) to_power n = (x,y) to_power n
proof
  defpred P[set] means for m holds m=$1 & m<= n implies (x,x\(x\y)) to_power m
  = (x,y) to_power m;
  now
    let k;
    assume
A1: for m holds m=k & m<= n implies (x,x\(x\y)) to_power m = (x,y) to_power m;
    let m;
A2: (x,x\(x\y)) to_power (k+1) =(x,x\(x\y)) to_power k\(x\(x\y)) by Th4;
    assume m=k+1 & m<=n;
    then k<=n by NAT_1:13;
    hence (x,x\(x\y)) to_power (k+1)=(x,y) to_power k \(x\(x\y)) by A1,A2
      .=(x\(x\(x\y)),y) to_power k by Th7
      .=(x\y,y) to_power k by BCIALG_1:8
      .=(x,y) to_power k\y by Th7
      .=(x,y) to_power (k+1) by Th4;
  end;
  then
A3: for k st P[k] holds P[k+1];
  (x,x\(x\y)) to_power 0 = x by Th1;
  then
A4: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis;
end;
