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 Th76:
  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 z being Element of X1, y being Element of Y st
  z in B1 & y in lim_filter(ProjMap1(f,z),cF2) holds
  y in Cl Int U
  proof
    assume that
A1: x in lim_filter(f,<.cF1,cF2.)) and
A2: <.cB1.) = cF1 and
A3: <.cB2.) = cF2;
    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;
A5: for y be Element of B1 holds f.:([:{y},B2:]) c= Int U
    proof
      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 thesis by A4;
    end;
A6: for z be Element of B1, y be Element of Y st
      y in lim_filter(ProjMap1(f,z),cF2)
    holds filter_image(ProjMap1(f,z),cF2) is proper Filter of BoolePoset [#]Y
      & Int U in filter_image(ProjMap1(f,z),cF2) &
      y is_a_cluster_point_of filter_image(ProjMap1(f,z),cF2),Y
    proof
      let z be Element of B1,y be Element of Y;
      assume
A7:   y in lim_filter(ProjMap1(f,z),cF2);
      filter_image(ProjMap1(f,z),cF2) is Filter of [#]Y by STRUCT_0:def 3;
      hence filter_image(ProjMap1(f,z),cF2) is
        proper Filter of BoolePoset [#]Y by Th17;
A8:   ProjMap1(f,z)"(Int U) is Subset of X2
      proof
        dom ProjMap1(f,z) = X2 by FUNCT_2:def 1;
        hence thesis by RELAT_1:132;
      end;
A9:   for z be Element of B1 holds ProjMap1(f,z).:B2 c= Int U
      proof
        let z be Element of B1;
        let t be object;
        assume t in ProjMap1(f,z).:B2;
        then consider u be object such that
        u in dom ProjMap1(f,z) and
A10:    u in B2 and
A11:    t = ProjMap1(f,z).u by FUNCT_1:def 6;
        ProjMap1(f,z).u = f.(z,u) by A10,MESFUNC9:def 6; then
A12:    t = f.([z,u]) by A11,BINOP_1:def 1;
        now
          [z,u] in [:X1,X2:] by A10,ZFMISC_1:def 2;
          hence [z,u] in dom f by FUNCT_2:def 1;
          z in {z} & u in B2 by TARSKI:def 1,A10;
          hence [z,u] in [:{z},B2:] by ZFMISC_1:def 2;
        end;
        then
A13:    t in f.:([:{z},B2:]) by A12,FUNCT_1:def 6;
        f.:([:{z},B2:]) c= Int U by A5;
        hence thesis by A13;
      end;
      thus Int U in filter_image(ProjMap1(f,z),cF2) &
        y is_a_cluster_point_of filter_image(ProjMap1(f,z),cF2),Y
      proof
        ProjMap1(f,z).:B2 c= Int U by A9;
        then B2 c= ProjMap1(f,z)"(Int U) by FUNCT_2:95;
        then ProjMap1(f,z)"(Int U) in cF2 by A3,A8,CARDFIL2:def 8;
        hence Int U in filter_image(ProjMap1(f,z),cF2);
        thus thesis by A7,Th19;
      end;
    end;
    for z be Element of B1, y be Element of Y st
      y in lim_filter(ProjMap1(f,z),cF2) holds y in Cl Int U
    proof
      let z be Element of B1, y be Element of Y;
      assume y in lim_filter(ProjMap1(f,z),cF2);
      then filter_image(ProjMap1(f,z),cF2) is
        proper Filter of BoolePoset [#]Y &
        Int U in filter_image(ProjMap1(f,z),cF2) &
        y is_a_cluster_point_of filter_image(ProjMap1(f,z),cF2),Y by A6;
      hence thesis by YELLOW19:25;
    end;
    hence thesis;
  end;
