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
  (0.X,x)to_power m=0.X implies(0.X,x)to_power (m*n)=0.X
proof
  defpred P[set] means for j being Nat holds j=$1 & j<=n implies (
  0.X,x)to_power (m*j)=0.X;
  assume
A1: (0.X,x)to_power m=0.X;
  now
    let k;
    assume
A2: for j st j=k & j<=n holds (0.X,x)to_power (m*j)=0.X;
    let j;
    assume j=k+1 & j<=n;
    then
A3: k<=n by NAT_1:13;
    (0.X,x)to_power (m*(k+1)) = (0.X,x)to_power (m*k+m)
      .= ((0.X,x)to_power (m*k),x)to_power m by Th10
      .= (0.X,x)to_power m by A2,A3;
    hence (0.X,x)to_power (m*(k+1))=0.X by A1;
  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;
