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 (z,y) to_power n <= (z,x) to_power n
proof
  defpred P[set] means for m being Nat holds m=$1 & m<=n implies (z
  ,y) to_power m <= (z,x) to_power m;
  assume
A1: x<=y;
A2: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A3: for m holds m=k & m<= n implies (z,y) to_power m <= (z,x) to_power m;
    ((z,x) to_power k)\y<=(z,x) to_power k\x by A1,BCIALG_1:5;
    then (z,x) to_power k\y<=(z,x) to_power (k+1) by Th4;
    then
A4: ((z,x) to_power k\y)\((z,x) to_power (k+1))=0.X;
    let m;
    assume that
A5: m=k+1 and
A6: m<=n;
    k<=n by A5,A6,NAT_1:13;
    then (z,y) to_power k <= (z,x) to_power k by A3;
    then (z,y) to_power k\y <= (z,x) to_power k\y by BCIALG_1:5;
    then (z,y) to_power (k+1)<= (z,x) to_power k\y by Th4;
    then ((z,y) to_power (k+1))\((z,x) to_power k\y)=0.X;
    then ((z,y) to_power (k+1))\((z,x) to_power (k+1))=0.X by A4,BCIALG_1:3;
    hence thesis by A5;
  end;
  z\z=0.X by BCIALG_1:def 5;
  then z<=z;
  then (z,y) to_power 0 <= z by Th1;
  then
A7: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A7,A2);
  hence thesis;
end;
