let T be non empty TopSpace; :: thesis: for s being Function of [:NAT,NAT:], the carrier of T

for x being Point of T holds

( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

let s be Function of [:NAT,NAT:], the carrier of T; :: thesis: for x being Point of T holds

( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

let x be Point of T; :: thesis: ( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

set F = filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).));

A1: ( filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

for x being Point of T holds

( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

let s be Function of [:NAT,NAT:], the carrier of T; :: thesis: for x being Point of T holds

( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

let x be Point of T; :: thesis: ( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

set F = filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).));

A1: ( filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )

proof

NeighborhoodSystem x c= filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))

end;

( filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x iff x in lim_filter (filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))) )
hereby :: thesis: ( ( for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A ) implies filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x )

assume A3:
for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A
; :: thesis: filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem xassume
filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x
; :: thesis: for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A

then NeighborhoodSystem x c= { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } by Th44;

then A2: { A where A is a_neighborhood of x : verum } c= { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } by YELLOW19:def 1;

thus for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A :: thesis: verum

end;then NeighborhoodSystem x c= { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } by Th44;

then A2: { A where A is a_neighborhood of x : verum } c= { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } by YELLOW19:def 1;

thus for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A :: thesis: verum

proof

let A be a_neighborhood of x; :: thesis: ex n being Nat st square-uparrow n c= s " A

A in { A where A is a_neighborhood of x : verum } ;

then A in { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } by A2;

then ex M being Subset of the carrier of T st

( A = M & ex n being Nat st square-uparrow n c= s " M ) ;

hence ex n being Nat st square-uparrow n c= s " A ; :: thesis: verum

end;A in { A where A is a_neighborhood of x : verum } ;

then A in { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } by A2;

then ex M being Subset of the carrier of T st

( A = M & ex n being Nat st square-uparrow n c= s " M ) ;

hence ex n being Nat st square-uparrow n c= s " A ; :: thesis: verum

NeighborhoodSystem x c= filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))

proof

hence
filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x
; :: thesis: verum
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in NeighborhoodSystem x or y in filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) )

assume y in NeighborhoodSystem x ; :: thesis: y in filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))

then y in { A where A is a_neighborhood of x : verum } by YELLOW19:def 1;

then consider A being a_neighborhood of x such that

A4: y = A ;

ex n being Nat st square-uparrow n c= s " A by A3;

then A in { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } ;

hence y in filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) by A4, Th44; :: thesis: verum

end;assume y in NeighborhoodSystem x ; :: thesis: y in filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))

then y in { A where A is a_neighborhood of x : verum } by YELLOW19:def 1;

then consider A being a_neighborhood of x such that

A4: y = A ;

ex n being Nat st square-uparrow n c= s " A by A3;

then A in { M where M is Subset of the carrier of T : ex n being Nat st square-uparrow n c= s " M } ;

hence y in filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) by A4, Th44; :: thesis: verum

proof

hence
( x in lim_filter (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) iff for A being a_neighborhood of x ex n being Nat st square-uparrow n c= s " A )
by A1; :: thesis: verum
thus
( filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x implies x in lim_filter (filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))) )
; :: thesis: ( x in lim_filter (filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))) implies filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x )

assume x in lim_filter (filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))) ; :: thesis: filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x

then ex y being Point of T st

( x = y & filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem y ) ;

hence filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x ; :: thesis: verum

end;assume x in lim_filter (filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).))) ; :: thesis: filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x

then ex y being Point of T st

( x = y & filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem y ) ;

hence filter_image (s,<.(Frechet_Filter NAT),(Frechet_Filter NAT).)) is_filter-finer_than NeighborhoodSystem x ; :: thesis: verum