reserve
  A,B,X for set,
  S for SigmaField of X;
reserve C for C_Measure of X;

theorem Th14:
  sigma_Meas(C) is sigma_Measure of sigma_Field(C)
proof
  reconsider M = sigma_Meas(C) as Measure of sigma_Field(C) by Th13;
  for F being Sep_Sequence of sigma_Field(C) holds M.(union rng F) <= SUM( M*F)
  proof
    let F be Sep_Sequence of sigma_Field(C);
    consider A being Subset of X such that
A1: A = union rng F;
A2: for k being object st k in NAT holds (M*F).k = (C*F).k
    proof
      let k be object;
      assume
A3:   k in NAT;
      then
A4:   (M*F).k = M.(F.k) by FUNCT_2:15;
A5:   F.k in sigma_Field(C) by A3,FUNCT_2:5;
      reconsider F as sequence of bool X by FUNCT_2:7;
      (C*F).k = C.(F.k) by A3,FUNCT_2:15;
      hence thesis by A4,A5,Def3;
    end;
    reconsider F9 = F as sequence of bool X by FUNCT_2:7;
    consider a,b being R_eal such that
    a = M.A and
A6: b = C.A;
    C*F9 is sequence of ExtREAL;
    then
A7: M*F = C*F by A2,FUNCT_2:12;
    reconsider F as sequence of bool X by FUNCT_2:7;
    b <= SUM(C*F) by A1,A6,Def1;
    hence thesis by A1,A6,A7,Def3;
  end;
  hence thesis by MEASURE3:14;
end;
