
theorem P136b:
  for k being Nat st k >= 1 holds
    PrimeDivisors (2 |^ k - 2) =
       {2} \/ PrimeDivisors (2 |^ (k-'1) - 1)
  proof
    let k be Nat;
    assume k >= 1; then
    PrimeDivisors (2 |^ k - 2) =
         PrimeDivisors (2 |^ (k-'1+1) - 2) by XREAL_1:235
      .= PrimeDivisors (2 |^ (k-'1) * 2 - 2) by NEWTON:6
      .= PrimeDivisors (2 * (2 |^ (k-'1) - 1))
      .= PrimeDivisors 2 \/ PrimeDivisors (2 |^ (k-'1) - 1)
         by DivisorsMulti
      .= {2} \/ PrimeDivisors (2 |^ (k-'1) - 1) by XPRIMES1:2,LemmaOne;
    hence thesis;
  end;
