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 Th8:
  for F being extreal-yielding FinSequence, r be Element of ExtREAL
  holds Product (F^<*r*>) = Product F * r
proof
  let F be extreal-yielding FinSequence, r be Element of ExtREAL;
A1: rng (F^<*r*>) c= ExtREAL by VALUED_0:def 2;
  rng F c= ExtREAL by VALUED_0:def 2;
  then reconsider Fr = F^<*r*>, Ff = F as FinSequence of ExtREAL by A1,
FINSEQ_1:def 4;
  reconsider Ff1=Ff as extreal-yielding FinSequence;
  Product (F^<*r*>) = multextreal $$ Fr by Def2;
  then Product (F^<*r*>) = multextreal.(multextreal $$ Ff,r) by FINSOP_1:4;
  then Product (F^<*r*>) = multextreal.(Product Ff1,r) by Def2;
  hence thesis by Def1;
end;
