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;
reserve Rseq for Function of [:NAT,NAT:],REAL;
reserve f for Function of [#]OrderedNAT,R^1,
        seq for Function of NAT,REAL;
reserve Y for non empty TopSpace,
        x for Point of Y,
        f for Function of [:X1,X2:],Y;

theorem
  x in lim_filter(f,<.cF1,cF2.)) & <.cB1.) = cF1 & <.cB2.) = cF2 implies
  for U being a_neighborhood of x st U is closed holds
  ex B1 being Element of cB1, B2 being Element of cB2 st
  for y being Element of B1 holds f.:([:{y},B2:]) c= Int(U)
  proof
    assume that
A1:  x in lim_filter(f,<.cF1,cF2.)) and
A2:  <.cB1.) = cF1 and
A3:  <.cB2.) = cF2;
    now
      let U be a_neighborhood of x;
      assume U is closed;
      then consider B1 be Element of cB1,B2 be Element of cB2 such that
A4:   f.:([:B1,B2:]) c= Int U by A1,A2,A3,Th75;
      take B1,B2;
      let y be Element of B1;
      [:{y},B2:] c= [:B1,B2:] by ZFMISC_1:95;
      then f.:([:{y},B2:]) c=f.:([:B1,B2:]) by RELAT_1:125;
      hence f.:([:{y},B2:]) c= Int U by A4;
    end;
    hence thesis;
  end;
