reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;

theorem Th17:
  a`|^n = (a|^n)`
proof
  defpred P[Nat] means a`|^$1 = (a|^$1)`;
A1: now
    let n;
    assume P[n];
    0.X in AtomSet(X);
    then a`|^(n+1) =(0.X)|^(n+1)\(a|^(n+1)) by Th15
      .= (a|^(n+1))` by Th7;
    hence P[n+1];
  end;
  a`|^0 =0.X by Def1
    .=(0.X)` by BCIALG_1:def 5;
  then
A2: P[0] by Def1;
  for n holds P[n] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
