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) = COM(P2M P)
proof
  set Y = COM(P);
  COM(Sigma,P) = COM(Sigma,P2M P) by Th27;
  then reconsider Y1=P2M(Y) as sigma_Measure of COM(Sigma,P2M(P));
  for B being set st B in Sigma for C being thin of P2M(P) holds Y1.(B \/
  C) = (P2M(P)).B
  proof
    let B be set such that
A1: B in Sigma;
    let C be thin of P2M(P);
    reconsider C1=C as thin of P by Th23;
    Y.(B \/ C1) = P.B by A1,Def8;
    hence thesis;
  end;
  hence thesis by MEASURE3:def 5;
end;
