reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,ExtREAL,
  E for Element of S;

theorem Th13:
  for k be Nat st 1<=k holds +infty|^k =+infty
proof
  defpred P[Nat] means +infty|^$1 = +infty;
A1: for k be non zero Nat st P[k] holds P[k+1]
  proof
    let k be non zero Nat;
    assume
A2: P[k];
    +infty|^(k+1)=(+infty|^k)*+infty by Th10;
    hence thesis by A2,XXREAL_3:def 5;
  end;
A3: P[1] by Th9;
  for k be non zero Nat holds P[k] from NAT_1:sch 10(A3,A1);
  hence thesis;
end;
