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=x implies (x,y) to_power n = x
proof
  defpred P[set] means for m holds m=$1 & m<=n implies (x,y) to_power m = x;
  assume
A1: x\y=x;
  now
    let k;
    assume
A2: for m holds m=k & m<= n implies (x,y) to_power m = x;
    let m;
A3: (x,y) to_power (k+1)=(x,y) to_power k \ y by Th4;
    assume m=k+1 & m<=n;
    then k<=n by NAT_1:13;
    hence (x,y) to_power (k+1)=x by A1,A2,A3;
  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;
