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 Th13:
  (0.X,x)to_power (n+m)=((0.X,x)to_power n)\((0.X,x)to_power m)`
proof
  defpred P[set] means for j being Nat holds j=$1 & j<=n implies (
  0.X,x)to_power (j+m)=((0.X,x)to_power j)\((0.X,x)to_power m)`;
  now
    let k;
    assume
A1: for j being Nat st j=k & j<=n holds (0.X,x)to_power (j+
    m)=((0.X,x)to_power j)\((0.X,x)to_power m)`;
    let j be Nat;
    assume j=k+1 & j<=n;
    then
A2: k<=n by NAT_1:13;
    (0.X,x)to_power (k+m+1)=((0.X,x)to_power (k+m))\x by Th4
      .=(((0.X,x)to_power k)\((0.X,x)to_power m)`)\x by A1,A2
      .=(((0.X,x)to_power k)\x)\((0.X,x)to_power m)` by BCIALG_1:7;
    hence
    (0.X,x)to_power ((k+1)+m) =((0.X,x)to_power (k+1))\((0.X,x)to_power m
    )` by Th4;
  end;
  then
A3: for k st P[k] holds P[k+1];
  (0.X,x)to_power (0+m) =(((0.X,x)to_power m)`)` by Th12;
  then
A4: P[0] by Th1;
  for n holds P[n] from NAT_1:sch 2(A4,A3);
  hence thesis;
end;
