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 Th33:
  for X being non empty set, F being Field_Subset of X, m being
Measure of F st m is completely-additive ex M be sigma_Measure of sigma F st M
  is_extension_of m & M = (sigma_Meas(C_Meas m))|(sigma F)
proof
  let X be non empty set, F be Field_Subset of X, m be Measure of F;
  assume
A1: m is completely-additive;
  set M = (sigma_Meas(C_Meas m))|(sigma F);
A2: F c= sigma_Field(C_Meas m) by Th20;
  then
A3: sigma F c= sigma_Field(C_Meas m) by PROB_1:def 9;
  then reconsider M as Function of sigma F,ExtREAL by FUNCT_2:32;
A4: for SS being Sep_Sequence of sigma F holds SUM(M*SS) = M.(union rng SS)
  proof
    let SS be Sep_Sequence of sigma F;
    reconsider SS9 = SS as Sep_Sequence of sigma_Field(C_Meas m) by A3,
FUNCT_2:7;
A5: rng SS c= sigma F by RELAT_1:def 19;
    M*SS = (sigma_Meas(C_Meas m))*((sigma F)|`SS) by MONOID_1:1
      .= (sigma_Meas(C_Meas m))*SS9 by A5,RELAT_1:94;
    then
A6: SUM(M*SS) = (sigma_Meas(C_Meas m)).(union rng SS9) by MEASURE1:def 6;
    union rng SS is Element of sigma F by MEASURE1:24;
    hence SUM(M*SS) = M.(union rng SS) by A6,FUNCT_1:49;
  end;
  M.{} = (sigma_Meas(C_Meas m)).{} by FUNCT_1:49,PROB_1:4
    .= 0 by VALUED_0:def 19;
  then reconsider M as sigma_Measure of sigma F by A4,MEASURE1:def 6
,MESFUNC5:15,VALUED_0:def 19;
  take M;
A7: F c= sigma F by PROB_1:def 9;
  for A be set st A in F holds M.A = m.A
  proof
    let A be set;
    assume
A8: A in F;
    then reconsider A9 = A as Subset of X;
    M.A = (sigma_Meas(C_Meas m)).A by A7,A8,FUNCT_1:49
      .= (C_Meas m).A9 by A2,A8,MEASURE4:def 3;
    hence M.A = m.A by A1,A8,Th18;
  end;
  hence M is_extension_of m & M = (sigma_Meas(C_Meas m))|(sigma F);
end;
