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 Th5:
  (x,0.X) to_power (n+1) = x
proof
  defpred P[set] means for m holds m=$1 & m<= n implies (x,0.X) to_power (m+1)
  = x;
  now
    let k;
    assume
A1: for m st m=k & m<= n holds (x,0.X) to_power (m+1) = x;
    let m;
    assume that
A2: m=k+1 and
A3: m<=n;
    (x,0.X) to_power (m+1)=((x,0.X) to_power (k+1))\0.X by A2,Th4;
    then
A4: (x,0.X) to_power (m+1)=(x,0.X) to_power (k+1) by BCIALG_1:2;
    k<=n by A2,A3,NAT_1:13;
    hence (x,0.X) to_power (m+1)= x by A1,A4;
  end;
  then
A5: for k st P[k] holds P[k+1];
  (x,0.X) to_power (0+1) = x \ 0.X by Th2;
  then
A6: P[0] by BCIALG_1:2;
  for n holds P[n] from NAT_1:sch 2(A6,A5);
  hence thesis;
end;
