reserve X for set;

theorem
  for S being SigmaField of X, M being sigma_Measure of S holds COM(M)
  is complete
proof
  let S be SigmaField of X, M be sigma_Measure of S;
  for A being Subset of X, B being set st B in COM(S,M) holds (A c= B & (
  COM(M)).B = 0. implies A in COM(S,M))
  proof
    let A be Subset of X;
    let B be set;
    assume
A1: B in COM(S,M);
    assume that
A2: A c= B and
A3: (COM(M)).B = 0.;
    ex B1 being set st (B1 in S & ex C1 being thin of M st A = B1 \/ C1)
    proof
      take {};
      consider B2 being set such that
A4:   B2 in S and
A5:   ex C2 being thin of M st B = B2 \/ C2 by A1,Def3;
A6:   M.B2 = 0. by A3,A4,A5,Def5;
      consider C2 being thin of M such that
A7:   B = B2 \/ C2 by A5;
      set C1 = (A /\ B2) \/ (A /\ C2);
      consider D2 being set such that
A8:   D2 in S and
A9:   C2 c= D2 and
A10:  M.D2 = 0. by Def2;
      set O = B2 \/ D2;
      A /\ C2 c= C2 by XBOOLE_1:17;
      then
A11:  A /\ B2 c= B2 & A /\ C2 c= D2 by A9,XBOOLE_1:17;
      ex O being set st O in S & C1 c= O & M.O = 0.
      proof
        reconsider B2,D2 as Element of S by A4,A8;
        reconsider O1 = O as Element of S by A4,A8,FINSUB_1:def 1;
        take O;
        M.(B2 \/ D2) <= 0. + 0. & 0. <= M.O1 by A6,A10,MEASURE1:33,def 2;
        hence thesis by A11,XBOOLE_1:13,XXREAL_0:1;
      end;
      then
A12:  C1 is thin of M by Def2;
      A = A /\ (B2 \/ C2) by A2,A7,XBOOLE_1:28
        .= {} \/ C1 by XBOOLE_1:23;
      hence thesis by A12,PROB_1:4;
    end;
    hence thesis by Def3;
  end;
  hence thesis;
end;
