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 V being Subset of Y st V is open & x in V holds
  ex B1 being Element of cB1, B2 being Element of cB2 st f.:([:B1,B2:]) c= V
  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 V be Subset of Y;
    assume that
A4: V is open and
A5: x in V;
    V in {M where M is Subset of Y: f"(M) in <.cF1,cF2.)}
      by A1,A4,A5,CARDFIL2:80,WAYBEL_7:def 5;
    then consider M be Subset of Y such that
A6: V = M and
A7: f"(M) in <.cF1,cF2.);
    <.cF1,cF2.) = <.[:cB1,cB2:].) by A2,A3,Def1;
    then consider B be Element of [:cB1,cB2:] such that
A8: B c= f"(M) by A7,CARDFIL2:def 8;
    B in [:cB1,cB2:];
    then consider B1 be Element of cB1, B2 be Element of cB2 such that
A9: B = [:B1,B2:];
    take B1,B2;
A10: f.:([:B1,B2:]) c= f.:(f"(M)) by A8,A9,RELAT_1:123;
    f.:(f"(M)) c= M by FUNCT_1:75;
    hence thesis by A6,A10;
  end;
