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 Th11:
  for x be Element of ExtREAL, y be Real, k be Nat
  st x=y holds x|^k = y|^k
proof
  let x be Element of ExtREAL, y be Real, k be Nat;
  defpred P[Nat] means x|^$1=y|^$1;
  assume
A1: x=y;
A2: for k be Nat st P[k] holds P[k+1]
  proof
    reconsider y1=y as Element of REAL by XREAL_0:def 1;
    let k be Nat;
    assume P[k];
    then (x|^k)*x = (y|^k)*y by A1,EXTREAL1:1;
    then (x|^k)*x = y|^(k+1) by NEWTON:6;
    hence thesis by Th10;
  end;
  x|^0 = 1. by Th6,FINSEQ_2:58;
  then
A3: P[0] by NEWTON:4;
  for k be Nat holds P[k] from NAT_1:sch 2(A3,A2);
  hence thesis;
end;
