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 Th10:
  ((x,y) to_power n,y) to_power m =(x,y) to_power (n+m)
proof
  defpred P[set] means for m1 being Nat holds m1=$1 & m1<= n
  implies ((x,y) to_power m1,y) to_power m =(x,y) to_power (m1+m);
  now
    let k;
    assume
A1: for m1 being Nat st m1=k & m1<=n holds((x,y) to_power
    m1,y) to_power m =(x,y) to_power (m1+m);
    let m1 be Nat;
    assume that
A2: m1=k+1 and
A3: m1<=n;
    k<=n by A2,A3,NAT_1:13;
    then ((x,y) to_power k,y) to_power m =(x,y) to_power (k+m) by A1;
    then (((x,y) to_power k,y) to_power m)\y=(x,y) to_power (k+m+1) by Th4;
    then (((x,y) to_power k)\y,y) to_power m=(x,y) to_power (k+m+1) by Th7;
    hence ((x,y) to_power m1,y) to_power m=(x,y) to_power (m1+m) by A2,Th4;
  end;
  then
A4: for k st P[k] holds P[k+1];
A5: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A5,A4);
  hence thesis;
end;
