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;
