
theorem P136a:
  for k being Nat st k >= 1 holds
    PrimeDivisors ((2 |^ k) * (2 |^ k - 2)) =
       {2} \/ PrimeDivisors (2 |^ (k-'1) - 1)
  proof
    let k be Nat;
    assume
A1: k >= 1; then
    2 |^ k >= 2 by PREPOWER:12; then
    2 |^ k - 2 >= 2 - 2 by XREAL_1:13; then
    2 |^ k - 2 >= 0; then
    PrimeDivisors ((2 |^ k) * (2 |^ k - 2)) =
        PrimeDivisors (2 |^ k) \/ PrimeDivisors (2 |^ k - 2)
          by DivisorsMulti
      .= {2} \/ PrimeDivisors (2 |^ k - 2) by A1,DivK,XPRIMES1:2
      .= {2} \/ PrimeDivisors (2 |^ (k-'1+1) - 2) by XREAL_1:235,A1
      .= {2} \/ PrimeDivisors (2 |^ (k-'1) * 2 - 2) by NEWTON:6
      .= {2} \/ PrimeDivisors (2 * (2 |^ (k-'1) - 1))
      .= {2} \/ (PrimeDivisors 2 \/ PrimeDivisors (2 |^ (k-'1) - 1))
         by DivisorsMulti
      .= {2} \/ ({2} \/ PrimeDivisors (2 |^ (k-'1) - 1))
         by LemmaOne,XPRIMES1:2
      .= {2} \/ {2} \/ PrimeDivisors (2 |^ (k-'1) - 1) by XBOOLE_1:4
      .= {2} \/ PrimeDivisors (2 |^ (k-'1) - 1);
    hence thesis;
  end;
