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\y) to_power n=((0.X,x) to_power n)\((0.X,y) to_power n)
proof
  defpred P[set] means for m holds m=$1 & m<=n implies (0.X,x\y) to_power m=((
  0.X,x) to_power m)\((0.X,y) to_power m);
A1: for k st P[k] holds P[k+1]
  proof
    let k;
    assume
A2: for m holds m=k & m<= n implies (0.X,x\y) to_power m=((0.X,x)
    to_power m)\((0.X,y) to_power m);
    let m;
    assume that
A3: m=k+1 and
A4: m<=n;
    k<=n by A3,A4,NAT_1:13;
    then
    (0.X,x\y) to_power k =((0.X,x) to_power k)\((0.X,y) to_power k) by A2;
    then
    (0.X,x\y) to_power (k+1) =(((0.X,x) to_power k)\((0.X,y) to_power k))
    \(x\y)by Th4
      .= ((0.X,x) to_power k)\(x\y)\((0.X,y) to_power k)by BCIALG_1:7
      .= (((x\y)`,x) to_power k)\((0.X,y) to_power k)by Th7
      .=((x`\y`,x) to_power k)\((0.X,y) to_power k)by BCIALG_1:9
      .=((x`,x) to_power k)\y`\((0.X,y) to_power k)by Th7
      .=((x`,x) to_power k)\((0.X,y) to_power k)\y` by BCIALG_1:7
      .=((0.X,x) to_power k)\x\((0.X,y) to_power k)\y` by Th7;
    then (0.X,x\y) to_power (k+1) =((0.X,x) to_power (k+1))\((0.X,y) to_power
    k)\y` by Th4
      .=((0.X,x) to_power (k+1))\y`\((0.X,y) to_power k)by BCIALG_1:7
      .=(((y`)`,x) to_power (k+1))\((0.X,y) to_power k)by Th7
      .=(((y`)`\((0.X,y) to_power k),x) to_power (k+1))by Th7
      .=(((0.X,y) to_power k)`\y`,x) to_power (k+1)by BCIALG_1:7
      .=((((0.X,y) to_power k)\y)`,x) to_power (k+1)by BCIALG_1:9
      .=(((0.X,y) to_power (k+1))`,x) to_power (k+1)by Th4;
    hence thesis by A3,Th7;
  end;
  (0.X)`=0.X by BCIALG_1:def 5;
  then (0.X,x\y) to_power 0=(0.X)` by Th1;
  then (0.X,x\y) to_power 0=((0.X,x) to_power 0)\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;
