reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];

theorem Th19:
  for T being non empty TopSpace, cF being Filter of the carrier of T st
  x in lim_filter cF holds x is_a_cluster_point_of cF,T
  proof
    let T be non empty TopSpace, cF be Filter of the carrier of T;
    assume
A1: x in lim_filter cF;
    now
      let A be Subset of T;
      assume that
A2:   A is open and
A3:   x in A;
A4:   A in cF by A1,A3,A2,CARDFIL2:80,WAYBEL_7:def 5;
      not {} in cF by CARD_FIL:def 1;
      hence for B be set st B in cF holds A meets B by A4,CARD_FIL:def 1;
    end;
    hence thesis by WAYBEL_7:def 4;
  end;
