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 Th35:
  for A being set holds (A in COM(Sigma,P) iff ex A1,A2 being set
  st A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 & P.(A2 \ A1) = 0)
proof
  let A be set;
  thus A in COM(Sigma,P) implies ex A1,A2 being set st A1 in Sigma & A2 in
  Sigma & A1 c= A & A c= A2 & P.(A2 \ A1) = 0
  proof
    assume A in COM(Sigma,P);
    then consider B being set such that
A1: B in Sigma and
A2: ex C being thin of P st A = B \/ C by Def5;
    consider C being thin of P such that
A3: A = B \/ C by A2;
    reconsider B as Event of Sigma by A1;
    consider D being set such that
A4: D in Sigma and
A5: C c= D and
A6: P.D = 0 by Def4;
    reconsider D as Event of Sigma by A4;
    reconsider E = D \/ B as Event of Sigma;
A7: P.(D \/ B) <= P.D + P.B by PROB_1:39;
    P.(E \ B) = P.(D \ B) by XBOOLE_1:40
      .= P.(D \/ B) - P.B by Th5;
    then P.(E \ B) <= 0 by A6,A7,XREAL_1:47;
    then
A8: P.(E \ B) = 0 by PROB_1:def 8;
    A c= E by A3,A5,XBOOLE_1:9;
    hence thesis by A2,A8,XBOOLE_1:7;
  end;
  thus (ex A1,A2 being set st A1 in Sigma & A2 in Sigma & A1 c= A & A c= A2 &
  P.(A2 \ A1) = 0) implies A in COM(Sigma,P)
  proof
    given A1,A2 being set such that
A9: A1 in Sigma and
A10: A2 in Sigma and
A11: A1 c= A and
A12: A c= A2 and
A13: P.(A2 \ A1) = 0;
    reconsider A2 as Element of Sigma by A10;
    reconsider A1 as Element of Sigma by A9;
    set C = A \ A1;
A14: C is thin of P
    proof
      reconsider D = A2 \ A1 as Element of Sigma;
A15:  C c= D
      proof
        let x be object;
        assume x in C;
        then x in A & not x in A1 by XBOOLE_0:def 5;
        hence thesis by A12,XBOOLE_0:def 5;
      end;
      then C c= Omega by XBOOLE_1:1;
      hence thesis by A13,A15,Def4;
    end;
    A = A1 \/ C by A11,XBOOLE_1:45;
    hence thesis by A14,Def5;
  end;
end;
