reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th24:
  for X be non empty set, F be SetSequence of X st F is
non-ascending for S be sequence of X st for n holds S.n in F.n holds rng S
  is finite implies meet F is non empty
proof
  let X be non empty set, F be SetSequence of X such that
A1: F is non-ascending;
  let S be sequence of X such that
A2: for n holds S.n in F.n;
A3: dom S=NAT by FUNCT_2:def 1;
  assume rng S is finite;
  then consider x being object such that
  x in rng S and
A4: S"{x} is infinite by A3,CARD_2:101;
  now
    let n be Nat;
    ex i st x in F.i & i >= n
    proof
      assume
A5:   for i st x in F.i holds i<n;
      S"{x} c= Segm n
      proof
        let y be object such that
A6:     y in S"{x};
        reconsider i=y as Nat by A6;
        S.i in {x} by A6,FUNCT_1:def 7;
        then
A7:     S.i=x by TARSKI:def 1;
        S.i in F.i by A2;
        then i<n by A5,A7;
        hence thesis by NAT_1:44;
      end;
      hence thesis by A4;
    end;
    then consider i such that
A8: x in F.i and
A9: i>=n;
    F.i c= F.n by A1,A9,PROB_1:def 4;
    hence x in F.n by A8;
  end;
  hence thesis by KURATO_0:3;
end;
