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 Th7:
  (x,y) to_power n\z = (x\z,y) to_power n
proof
  defpred P[set] means for m holds m=$1 & m<= n implies (x,y) to_power m\z = (
  x\z,y) to_power m;
  now
    let k;
    assume
A1: for m st m=k & m<= n holds (x,y) to_power m\z = (x\z,y) to_power m;
    let m;
    assume that
A2: m=k+1 and
A3: m<=n;
    (x,y) to_power m\z=((x,y) to_power k\y)\z by A2,Th4;
    then
A4: (x,y) to_power m\z=((x,y) to_power k\z)\y by BCIALG_1:7;
    k<=n by A2,A3,NAT_1:13;
    then (x,y) to_power m\z= ((x\z,y) to_power k)\y by A1,A4;
    hence (x,y) to_power m\z= (x\z,y) to_power (k+1) by Th4;
  end;
  then
A5: for k st P[k] holds P[k+1];
  (x,y) to_power 0\z = x\z by Th1;
  then
A6: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A6,A5);
  hence thesis;
end;
