theorem Th75:
  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
  f.:([:B1,B2:]) c= Int(U)
  proof
    assume that
A1: x in lim_filter(f,<.cF1,cF2.)) and
A2: <.cB1.) = cF1 and
A3: <.cB2.) = cF2;
    reconsider FF = filter_image(f,<.cF1,cF2.)) as Filter of the carrier of Y;
    let U be a_neighborhood of x;
    assume U is closed;
    x in Int(U) by CONNSP_2:def 1;
    then Int(U) in {M where M is Subset of Y: f"(M) in <.cF1,cF2.)}
      by A1,CARDFIL2:80,WAYBEL_7:def 5;
    then consider M be Subset of Y such that
A4: Int(U) = M and
A5: f"(M) in <.cF1,cF2.);
    <.cF1,cF2.) = <.[:cB1,cB2:].) by A2,A3,Def1;
    then consider B be Element of [:cB1,cB2:] such that
A6: B c= f"(M) by A5,CARDFIL2:def 8;
    B in [:cB1,cB2:];
    then consider B1 be Element of cB1, B2 be Element of cB2 such that
A7: B = [:B1,B2:];
    take B1,B2;
A8: f.:([:B1,B2:]) c= f.:(f"(M)) by A6,A7,RELAT_1:123;
    f.:(f"(M)) c= M by FUNCT_1:75;
    hence thesis by A4,A8;
  end;
