reserve X for set,
  F for Field_Subset of X,
  M for Measure of F,
  A,B for Subset of X,
  Sets for SetSequence of X,
  seq,seq1,seq2 for ExtREAL_sequence,
  n,k for Nat;
reserve FSets for Set_Sequence of F,
  CA for Covering of A,F;
reserve Cvr for Covering of Sets,F;
reserve C for C_Measure of X;

theorem Th21:
  for X be set, F be Field_Subset of X, FSets be Set_Sequence of F
  , M be Function of F,ExtREAL holds M * FSets is ExtREAL_sequence
proof
  let X be set;
  let F be Field_Subset of X;
  let FSets be Set_Sequence of F;
  let M be Function of F,ExtREAL;
  now
    let y be object;
    assume y in rng FSets;
    then ex x be object st x in NAT & FSets.x = y by FUNCT_2:11;
    hence y in F by Def2;
  end;
  then rng FSets c= F;
  then rng FSets c= dom M by FUNCT_2:def 1;
  then dom(M * FSets) = dom FSets by RELAT_1:27;
  then dom(M * FSets) = NAT by FUNCT_2:def 1;
  hence thesis by FUNCT_2:def 1;
end;
