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,y) to_power n,z) to_power m =((x,z) to_power m,y)to_power n
proof
  defpred P[set] means for m1 being Nat holds m1=$1 & m1<= n
  implies ((x,y) to_power m1,z) to_power m =((x,z) to_power m,y)to_power m1;
  now
    let k;
    assume
A1: for m1 being Nat st m1=k & m1<=n holds ((x,y) to_power
    m1,z) to_power m =((x,z) to_power m,y)to_power m1;
    let m1 be Nat;
    assume m1=k+1 & m1<=n;
    then k<=n by NAT_1:13;
    then
    (((x,y) to_power k,z) to_power m)\y =((x,z) to_power m,y)to_power k \
    y by A1
      .=((x,z) to_power m,y)to_power (k+1) by Th4;
    then
    ((x,z) to_power m,y)to_power (k+1) =((x,y) to_power k\y,z) to_power m
    by Th7;
    hence
    ((x,y) to_power (k+1),z) to_power m =((x,z) to_power m,y)to_power (k+
    1) by Th4;
  end;
  then
A2: for k st P[k] holds P[k+1];
  ((x,y) to_power 0,z)to_power m =(x,z)to_power m by Th1;
  then
A3: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
