
theorem
  for R being well-unital left_zeroed right_zeroed add-cancelable
  associative distributive non empty doubleLoopStr, I being right-ideal non
  empty Subset of R, a being Element of I, n being Element of NAT st n <> 0
  holds a|^n in I
proof
  let R be well-unital left_zeroed right_zeroed add-cancelable associative
distributive non empty doubleLoopStr, I be right-ideal non empty Subset of R
  , a be Element of I, n be Element of NAT;
  defpred P[Nat] means a|^$1 in I;
  assume
A1: n <> 0;
A2: for n being Nat st 1 <= n holds P[n] implies P[n+1]
  proof
    let n be Nat;
    assume 1 <= n;
A3: a|^(n+1) = (a|^n)*(a|^1) by BINOM:10;
    assume a|^n in I;
    hence thesis by A3,Def3;
  end;
  a|^1 = a by BINOM:8;
  then
A4: P[1];
  for n being Nat st 1 <= n holds P[n] from NAT_1:sch 8(A4,A2);
  hence thesis by A1,NAT_1:14;
end;
