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
  A1 is non-descending implies for n holds A1.(n+1) = (
  Partial_Diff_Union A1).(n+1) \/ A1.n
proof
  assume
A1: A1 is non-descending;
  thus for n holds A1.(n+1) = (Partial_Diff_Union A1).(n+1) \/ A1.n
  proof
    let n;
A2: A1.n c= A1.(n+1) by A1,PROB_2:7;
    thus (Partial_Diff_Union A1).(n+1) \/ A1.n = (A1.(n+1) \ (Partial_Union A1
    ).n) \/ A1.n by PROB_3:def 3
      .= (A1.(n+1) \ A1.n) \/ A1.n by A1,Lm2
      .= A1.(n+1) \/ A1.n by XBOOLE_1:39
      .= A1.(n+1) by A2,XBOOLE_1:12;
  end;
end;
