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

for x being Point of T

for cB being basis of (BOOL2F (NeighborhoodSystem x)) holds

( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

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

for cB being basis of (BOOL2F (NeighborhoodSystem x)) holds

( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

let x be Point of T; :: thesis: for cB being basis of (BOOL2F (NeighborhoodSystem x)) holds

( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

let cB be basis of (BOOL2F (NeighborhoodSystem x)); :: thesis: ( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

for x being Point of T

for cB being basis of (BOOL2F (NeighborhoodSystem x)) holds

( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

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

for cB being basis of (BOOL2F (NeighborhoodSystem x)) holds

( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

let x be Point of T; :: thesis: for cB being basis of (BOOL2F (NeighborhoodSystem x)) holds

( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

let cB be basis of (BOOL2F (NeighborhoodSystem x)); :: thesis: ( x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) iff for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A )

hereby :: thesis: ( ( for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A ) implies x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) )

assume A2:
for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A
; :: thesis: x in lim_filter (s,(Frechet_Filter [:NAT,NAT:]))assume A1:
x in lim_filter (s,(Frechet_Filter [:NAT,NAT:]))
; :: thesis: for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ A

end;now :: thesis: for A being a_neighborhood of x holds [:NAT,NAT:] \ (s " A) is finite

hence
x in lim_filter (s,(Frechet_Filter [:NAT,NAT:]))
by Th46; :: thesis: verumlet A be a_neighborhood of x; :: thesis: [:NAT,NAT:] \ (s " A) is finite

A3: A is Element of BOOL2F (NeighborhoodSystem x) by YELLOW19:2;

cB is filter_basis ;

then consider B being Element of cB such that

A4: B c= A by A3;

ex C being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ C by A2;

hence [:NAT,NAT:] \ (s " A) is finite by A4, RELAT_1:145, Th1; :: thesis: verum

end;A3: A is Element of BOOL2F (NeighborhoodSystem x) by YELLOW19:2;

cB is filter_basis ;

then consider B being Element of cB such that

A4: B c= A by A3;

ex C being finite Subset of [:NAT,NAT:] st s " B = [:NAT,NAT:] \ C by A2;

hence [:NAT,NAT:] \ (s " A) is finite by A4, RELAT_1:145, Th1; :: thesis: verum