reserve n,m,k for Element of NAT,
  x,X for set,
  A1 for SetSequence of X,
  Si for SigmaField of X,
  XSeq for SetSequence of Si;
reserve Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq for SetSequence of Sigma,
  P for Probability of Sigma;

theorem
  COM(P) is complete
proof
  for A being Subset of Omega for B being set st B in COM(Sigma,P) & A c=
  B & (COM P).B = 0 holds A in COM(Sigma,P)
  proof
    let A be Subset of Omega;
    let B be set;
    assume
A1: B in COM(Sigma,P);
    assume that
A2: A c= B and
A3: (COM P).B = 0;
    ex B1 being set st (B1 in Sigma & ex C1 being thin of P st A = B1 \/ C1 )
    proof
      take {};
      consider B2 being set such that
A4:   B2 in Sigma and
A5:   ex C2 being thin of P st B = B2 \/ C2 by A1,Def5;
A6:   P.B2 = 0 by A3,A4,A5,Def8;
      consider C2 being thin of P such that
A7:   B = B2 \/ C2 by A5;
      set C1 = (A /\ B2) \/ (A /\ C2);
      consider D2 being set such that
A8:   D2 in Sigma and
A9:   C2 c= D2 and
A10:  P.D2 = 0 by Def4;
      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 Sigma & C1 c= O & P.O = 0
      proof
        reconsider B2,D2 as Element of Sigma by A4,A8;
        take O;
        P.(B2 \/ D2) <= 0 + 0 by A6,A10,PROB_1:39;
        hence thesis by A11,PROB_1:def 8,XBOOLE_1:13;
      end;
      then
A12:  C1 is thin of P by Def4;
      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 Def5;
  end;
  hence thesis;
end;
