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 A being Element of COM(Sigma,P) holds ProbPart(A) = MeasPart(
  P_COM2M_COM(A))
proof
  let A be Element of COM(Sigma,P);
A1: MeasPart(P_COM2M_COM(A)) c= ProbPart(A)
  proof
    let x be object such that
A2: x in MeasPart(P_COM2M_COM A);
   reconsider xx=x as set by TARSKI:1;
    P_COM2M_COM(A) \ xx is thin of P2M(P) by A2,MEASURE3:def 4;
    then
A3: A \ xx is thin of P by Th23;
    x in Sigma & xx c= P_COM2M_COM(A) by A2,MEASURE3:def 4;
    hence thesis by A3,Def7;
  end;
  ProbPart(A) c= MeasPart(P_COM2M_COM(A))
  proof
    let x be object such that
A4: x in ProbPart(A);
   reconsider xx=x as set by TARSKI:1;
    A \ xx is thin of P by A4,Def7;
    then
A5: P_COM2M_COM(A) \ xx is thin of P2M(P) by Th23;
    x in Sigma & xx c= A by A4,Def7;
    hence thesis by A5,MEASURE3:def 4;
  end;
  hence thesis by A1;
end;
