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 Th69:
  Z is MonotoneClass of X iff Z c= bool X & for A1 being
  SetSequence of X st A1 is monotone & rng A1 c= Z holds lim A1 in Z
proof
  thus Z is MonotoneClass of X implies Z c= bool X & for A1 being SetSequence
  of X st A1 is monotone & rng A1 c= Z holds lim A1 in Z
  proof
    assume
A1: Z is MonotoneClass of X;
    then reconsider Z as Subset-Family of X;
    for A1 being SetSequence of X st A1 is monotone & rng A1 c= Z holds
    lim A1 in Z
    proof
      let A1 be SetSequence of X;
      assume that
A2:   A1 is monotone and
A3:   rng A1 c= Z;
      per cases by A2,SETLIM_1:def 1;
      suppose
        A1 is non-descending;
        hence thesis by A1,A3,Th66;
      end;
      suppose
        A1 is non-ascending;
        hence thesis by A1,A3,Th67;
      end;
    end;
    hence thesis;
  end;
  assume that
A4: Z c= bool X and
A5: for A1 being SetSequence of X st A1 is monotone & rng A1 c= Z holds
  lim A1 in Z;
  reconsider Z as Subset-Family of X by A4;
A6: for A1 being SetSequence of X st A1 is non-ascending & rng A1 c= Z
  holds lim A1 in Z
  proof
    let A1 be SetSequence of X;
    assume
A7: A1 is non-ascending & rng A1 c= Z;
    A1 is monotone & rng A1 c= Z implies lim A1 in Z by A5;
    hence thesis by A7,SETLIM_1:def 1;
  end;
  for A1 being SetSequence of X st A1 is non-descending & rng A1 c= Z
  holds lim A1 in Z
  proof
    let A1 be SetSequence of X;
    assume
A8: A1 is non-descending & rng A1 c= Z;
    A1 is monotone & rng A1 c= Z implies lim A1 in Z by A5;
    hence thesis by A8,SETLIM_1:def 1;
  end;
  hence thesis by A6,Th66,Th67;
end;
