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 Th23:
  for Y being Subset of Omega holds Y is thin of P iff Y is thin of P2M(P)
proof
  let Y be Subset of Omega;
  hereby
    assume Y is thin of P;
    then ex A be set st A in Sigma & Y c= A & P.A = 0 by Def4;
    hence Y is thin of P2M(P) by MEASURE3:def 2;
  end;
  assume Y is thin of P2M(P);
  then ex B be set st B in Sigma & Y c= B & (P2M(P)).B = 0. by MEASURE3:def 2;
  hence thesis by Def4;
end;
