theorem Th39:
  cF4 is_filter-finer_than cF3 implies lim_filter cF3 c= lim_filter cF4
  proof
    assume
A1: cF4 is_filter-finer_than cF3;
    let x be object;
    assume x in lim_filter cF3;
    then consider y be Point of T such that
A2: x = y and
A3: cF3 is_filter-finer_than NeighborhoodSystem y;
    NeighborhoodSystem y c= cF3 & cF3 c= cF4 by A1,A3;
    then NeighborhoodSystem y c= cF4;
    then cF4 is_filter-finer_than NeighborhoodSystem y;
    hence thesis by A2;
  end;
