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 Th4:
  (x,y) to_power (n+1) = (x,y) to_power n \ y
proof
A1: n<n+1 by NAT_1:3,XREAL_1:29;
  consider g such that
A2: (x,y) to_power n = g.n and
A3: g.0 = x and
A4: for j st j < n holds g.(j + 1) = g.j \ y by Def1;
  consider f such that
A5: (x,y) to_power (n+1) = f.(n+1) and
A6: f.0 = x and
A7: for j st j < n+1 holds f.(j + 1) = f.j \ y by Def1;
  defpred P[set] means for m holds m=$1 & m <= n implies f.m=g.m;
  now
    let k;
    assume
A8: for m st m=k & m<=n holds f.m=g.m;
    let m;
    assume that
A9: m=k+1 and
A10: m<=n;
    k +1 <= n+1 by A9,A10,NAT_1:13;
    then k<n+1 by NAT_1:13;
    then
A11: f.(k + 1) = f.k \y by A7;
A12: k<n by A9,A10,NAT_1:13;
    then g.(k + 1) = g.k \y by A4;
    hence f.m=g.m by A8,A9,A12,A11;
  end;
  then
A13: for k st P[k] holds P[k+1];
A14: P[0] by A6,A3;
  for n holds P[n] from NAT_1:sch 2(A14,A13);
  then f.n = g.n;
  hence thesis by A5,A7,A2,A1;
end;
