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 Th12:
  for x be Element of ExtREAL, k be Nat st 0 <=x holds 0 <= x|^k
proof
  let x be Element of ExtREAL, k be Nat;
  defpred P[Nat] means 0 <= x|^$1;
  assume
A1: 0 <=x;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    assume
A3: P[k];
    x|^(k+1)=(x|^k)*x by Th10;
    hence thesis by A1,A3;
  end;
A4: P[0] by Th6,FINSEQ_2:58;
  for k be Nat holds P[k] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
