let X1, X2 be non empty set ; :: thesis:
let cB1 be filter_base of X1; :: thesis:
let cB2 be filter_base of X2; :: thesis:
let cF1 be Filter of X1; :: thesis:
let cF2 be Filter of X2; :: thesis:
let Y be non empty TopSpace; :: thesis:
let x be Point of Y; :: thesis:
let f be Function of [:X1,X2:],Y; :: thesis:
assume that
A1: x in lim_filter (f,<.cF1,cF2.)) and
A2: <.cB1.) = cF1 and
A3: <.cB2.) = cF2 ; :: thesis:
let U be a_neighborhood of x; :: thesis:
assume U is closed ; :: thesis:
then consider B1 being Element of cB1, B2 being Element of cB2 such that
A4: f .: [:B1,B2:] c= Int U by A1, A2, A3, Th75;
take B1 ; :: thesis:
take B2 ; :: thesis:
A5: for y being Element of B1 holds f .: [:{y},B2:] c= Int U

let y be Element of B1; :: thesis:
[:{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; :: thesis:

end;

A6: for z being Element of B1
for y being 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 )

let z be Element of B1; :: thesis:
let y be Element of Y; :: thesis:
assume A7: y in lim_filter ((ProjMap1 (f,z)),cF2) ; :: thesis:
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; :: thesis:
A8: (ProjMap1 (f,z)) " (Int U) is Subset of X2

dom (ProjMap1 (f,z)) = X2 by FUNCT_2:def 1;
hence (ProjMap1 (f,z)) " (Int U) is Subset of X2 by RELAT_1:132; :: thesis:

end;

A9: for z being Element of B1 holds (ProjMap1 (f,z)) .: B2 c= Int U

let z be Element of B1; :: thesis:
let t be object ; :: according to TARSKI:def 3 :: thesis:
assume t in (ProjMap1 (f,z)) .: B2 ; :: thesis:
then consider u being 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;

[z,u] in [:X1,X2:] by A10, ZFMISC_1:def 2;
hence [z,u] in dom f by FUNCT_2:def 1; :: thesis:
( z in {z} & u in B2 ) by TARSKI:def 1, A10;
hence [z,u] in [:{z},B2:] by ZFMISC_1:def 2; :: thesis:

end;

then A13: t in f .: [:{z},B2:] by A12, FUNCT_1:def 6;
f .: [:{z},B2:] c= Int U by A5;
hence t in Int U by A13; :: thesis:

end;

thus ( Int U in filter_image ((ProjMap1 (f,z)),cF2) & y is_a_cluster_point_of filter_image ((ProjMap1 (f,z)),cF2),Y ) :: thesis:

(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) ; :: thesis:
thus y is_a_cluster_point_of filter_image ((ProjMap1 (f,z)),cF2),Y by A7, Th19; :: thesis:

end;

end;

for z being Element of B1
for y being Element of Y st y in lim_filter ((ProjMap1 (f,z)),cF2) holds
y in Cl (Int U)

let z be Element of B1; :: thesis:
let y be Element of Y; :: thesis:
assume y in lim_filter ((ProjMap1 (f,z)),cF2) ; :: thesis:
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 y in Cl (Int U) by YELLOW19:25; :: thesis:

end;

hence for z being Element of X1
for y being Element of Y st z in B1 & y in lim_filter ((ProjMap1 (f,z)),cF2) holds
y in Cl (Int U) ; :: thesis: