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), B being set st B in ProbPart(A)
  holds P.B = (COM P).A
proof
  let A be Element of COM(Sigma,P), B be set such that
A1: B in ProbPart(A);
  reconsider C = A \ B as thin of P by A1,Def7;
A2: B in Sigma by A1,Def7;
  B c= A by A1,Def7;
  then
A3: A = B \/ C by XBOOLE_1:45;
  Sigma c= COM(Sigma,P) by Th28;
  then reconsider B as Event of COM(Sigma,P) by A2;
  reconsider A as Event of COM(Sigma,P);
  B misses C by XBOOLE_1:79;
  then
A4: (COM P).A = (COM P).B + (COM P).C by A3,PROB_1:def 8
    .= (COM P).B + 0 by Th40
    .= (COM P).B;
  reconsider B as Event of Sigma by A1,Def7;
  (COM P).A = P.B by A4,Th39;
  hence thesis;
end;
