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:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;
reserve T for non empty TopSpace,
        s for Function of [:NAT,NAT:], the carrier of T,
        M for Subset of the carrier of T;
reserve cF3,cF4 for Filter of the carrier of T;

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;
