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;

theorem Th18:
  M is completely-additive implies for A be set st A in F holds M.
  A = (C_Meas M).A
proof
  assume
A1: M is completely-additive;
  let A be set;
  assume
A2: A in F;
  then reconsider A9 = A as Subset of X;
  for x be ExtReal st x in Svc(M,A9) holds M.A <= x
  proof
    let x be ExtReal;
    assume x in Svc(M,A9);
    then consider Aseq be Covering of A9,F such that
A3: x = SUM vol(M,Aseq) by Def7;
    consider Bseq being Sep_Sequence of F such that
A4: A = union rng Bseq and
A5: for n be Nat holds Bseq.n c= Aseq.n by A2,Th17;
    for n being Element of NAT holds (M*Bseq).n <= (vol(M,Aseq)).n
    proof
      let n be Element of NAT;
      M.(Bseq.n) <= M.(Aseq.n) by A5,MEASURE1:8;
      then (M*Bseq).n <= M.(Aseq.n) by FUNCT_2:15;
      hence thesis by Def5;
    end;
    then SUM(M*Bseq) <= SUM vol(M,Aseq) by SUPINF_2:43;
    hence M.A <= x by A1,A2,A3,A4;
  end;
  then M.A is LowerBound of Svc(M,A9) by XXREAL_2:def 2;
  then M.A <= inf Svc(M,A9) by XXREAL_2:def 4;
  then
A6: M.A <= (C_Meas M).A9 by Def8;
  (C_Meas M).A <= M.A by A2,Th9;
  hence M.A = (C_Meas M).A by A6,XXREAL_0:1;
end;
