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
  for C being non empty Subset-Family of Omega holds (for A being set
holds (A in C iff ex A1,A2 being set st A1 in Sigma & A2 in Sigma & A1 c= A & A
  c= A2 & P.(A2 \ A1) = 0)) implies C = COM(Sigma,P)
proof
  let C be non empty Subset-Family of Omega;
  assume
A1: for A being set holds (A in C iff ex A1,A2 being set st A1 in Sigma
  & A2 in Sigma & A1 c= A & A c= A2 & P.(A2 \ A1) = 0);
  now
    let A be object;
     reconsider AA=A as set by TARSKI:1;
    A in C iff ex A1,A2 being set st A1 in Sigma & A2 in Sigma & A1 c= AA &
    AA c= A2 & P.(A2 \ A1) = 0 by A1;
    hence A in C iff A in COM(Sigma,P) by Th35;
  end;
  hence thesis by TARSKI:2;
end;
