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 Th10:
  for x be Element of ExtREAL, k be Nat holds x|^(k+1) = x|^k*x
proof
  let x be Element of ExtREAL;
  defpred P[Nat] means x|^($1+1) = x|^$1*x;
A1: for k be Nat st P[k] holds P[k+(1 qua Complex)]
  proof
    let k be Nat;
    assume x|^(k+1) = x|^k*x;
    reconsider k1=k+1 as Element of NAT;
    Product((k1+1) |-> x) = Product((k1 |-> x) ^ <*x*>) by FINSEQ_2:60;
    hence thesis by Th8;
  end;
  x|^(0+1) = Product(<*x*>) by FINSEQ_2:59;
  then x|^(0+1) = x by Th7;
  then x|^(0+1) =1. * x by XXREAL_3:81;
  then
A2: P[0] by Th6,FINSEQ_2:58;
  for k be Nat holds P[k] from NAT_1:sch 2(A2,A1);
  hence thesis;
end;
