
theorem
  for z be Nat st 2 <= z holds
  for k be Nat
  ex N be Nat st
  for x be Nat st N <= x holds 1/ (z to_power x) < 1/ (x to_power k)
  proof
    let z be Nat;
    assume P0: 2 <=z;
    let k be Nat;
    consider N be Nat such that
    P1: for x be Nat st N <= x
    holds 1/ (2 to_power x) < 1/ (x to_power k) by L2;
    take N;
    now let x be Nat;
      assume N <=x; then
      E1:1/ (2 to_power x) < 1/ (x to_power k) by P1;
      P3: 0 < 2 to_power x by POWER:34;
      2 to_power x <= z to_power x
      proof
        now per cases by P0,XXREAL_0:1;
          case 2 = z;
            hence thesis;
          end;
          case LL1: 2 < z;
            now per cases;
              case LL2: x= 0;then
                2 to_power x = 1 by POWER:24
                .= z to_power x by POWER:24,LL2;
                hence thesis;
              end;
              case 0 < x;
                hence thesis by POWER:37,LL1;
              end;
            end;
            hence thesis;
          end;
        end;
        hence thesis;
      end;
      then 1/ (z to_power x) <= 1/ (2 to_power x) by P3,XREAL_1:118;
      hence 1/ (z to_power x) < 1/ (x to_power k) by E1,XXREAL_0:2;
    end;
    hence thesis;
  end;
