reserve n,m,k,i for Nat,
  g,s,t,p for Real,
  x,y,z for object, X,Y,Z for set,
  A1 for SetSequence of X,
  F1 for FinSequence of bool X,
  RFin for real-valued FinSequence,
  Si for SigmaField of X,
  XSeq,YSeq for SetSequence of Si,
  Omega for non empty set,
  Sigma for SigmaField of Omega,
  ASeq,BSeq for SetSequence of Sigma,
  P for Probability of Sigma;
reserve FSeq for FinSequence of Sigma;

theorem Th66:
  for IT be Subset-Family of X holds IT is non-decreasing-closed
iff for A1 being SetSequence of X st A1 is non-descending & rng A1 c= IT holds
  lim A1 in IT
proof
  let IT be Subset-Family of X;
  thus IT is non-decreasing-closed implies for A1 being SetSequence of X st A1
  is non-descending & rng A1 c= IT holds lim A1 in IT
  proof
    assume
A1: IT is non-decreasing-closed;
    now
      let A1 be SetSequence of X;
      assume that
A2:   A1 is non-descending and
A3:   rng A1 c= IT;
      Union A1 in IT by A1,A2,A3;
      hence lim A1 in IT by A2,SETLIM_1:63;
    end;
    hence thesis;
  end;
  assume
A4: for A1 being SetSequence of X st A1 is non-descending & rng A1 c= IT
  holds lim A1 in IT;
  for A1 being SetSequence of X st A1 is non-descending & rng A1 c= IT
  holds Union A1 in IT
  proof
    let A1 be SetSequence of X;
    assume that
A5: A1 is non-descending and
A6: rng A1 c= IT;
    lim A1 in IT by A4,A5,A6;
    hence thesis by A5,SETLIM_1:63;
  end;
  hence thesis;
end;
