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 implies (x,z) to_power n <= (y,z) to_power n
proof
  defpred P[set] means for m being Nat holds m=$1 & m<=n implies (x
  ,z) to_power m <= (y,z) to_power m;
A1: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A2: for m being Nat holds m=k & m<= n implies (x,z)
    to_power m <= (y,z) to_power m;
    let m be Nat;
    assume that
A3: m=k+1 and
A4: m<=n;
    k<=n by A3,A4,NAT_1:13;
    then (x,z) to_power k <= (y,z) to_power k by A2;
    then (x,z) to_power k\z<=(y,z) to_power k \ z by BCIALG_1:5;
    then (x,z) to_power (k+1)<=(y,z) to_power k\z by Th4;
    hence thesis by A3,Th4;
  end;
  assume x<=y;
  then (x,z) to_power 0 <= y by Th1;
  then
A5: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
