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 for A1,A2 being set st A1 in
  ProbPart(A) & A2 in ProbPart(A) holds P.A1 = P.A2
proof
  let A be Element of COM(Sigma,P);
  let A1,A2 be set such that
A1: A1 in ProbPart(A) and
A2: A2 in ProbPart(A);
  reconsider C1 = A \ A1, C2 = A \ A2 as thin of P by A1,A2,Def7;
A3: A2 c= A by A2,Def7;
  A1 c= A by A1,Def7;
  then
A4: A1 \/ C1 = A by XBOOLE_1:45
    .= A2 \/ C2 by A3,XBOOLE_1:45;
  A1 in Sigma & A2 in Sigma by A1,A2,Def7;
  hence thesis by A4,Th25;
end;
