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,x\y) to_power n),y\x) to_power n <= x
proof
  defpred P[set] means for m being Nat holds m=$1 & m<=n implies ((
  (x,x\y) to_power m),y\x) to_power m <= x;
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,x\y)
    to_power m),y\x) to_power m <= x;
    let m be Nat;
    assume that
A3: m=k+1 and
A4: m<=n;
    k<=n by A3,A4,NAT_1:13;
    then (((x,x\y) to_power k),y\x) to_power k <= x by A2;
    then ((((x,x\y) to_power k),y\x) to_power k)\x=0.X;
    then ((((x,x\y) to_power k)\x,y\x) to_power k)\(y\x)=(y\x)`by Th7;
    then ((((x,x\y) to_power k)\x,y\x)to_power (k+1))\(x\y) =(y\x)`\(x\y)by Th4
;
    then (((x,x\y) to_power k)\x\(x\y),y\x) to_power (k+1)=(y\x)`\(x\y)by Th7;
    then
(((x,x\y) to_power k)\(x\y)\x,y\x) to_power (k+1)=(y\x)`\(x\y) by BCIALG_1:7;
    then (((x,x\y) to_power (k+1))\x,y\x) to_power (k+1)=(y\x)`\(x\y)by Th4;
    then (((x,x\y) to_power (k+1))\x,y\x) to_power (k+1) =(y\y)\(y\x)\(x\y)by
BCIALG_1:def 5;
    then (((x,x\y) to_power (k+1))\x,y\x) to_power (k+1)=0.X by BCIALG_1:1;
    then (((x,x\y) to_power (k+1),y\x) to_power (k+1))\x=0.X by Th7;
    hence thesis by A3;
  end;
  x\x=0.X by BCIALG_1:def 5;
  then x<=x;
  then (x,y\x) to_power 0<=x by Th1;
  then
A5: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A5,A1);
  hence thesis;
end;
