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 .: ([:NAT,NAT:] \ A) c= B )
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 .: ([:NAT,NAT:] \ A) c= B )
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 .: ([:NAT,NAT:] \ A) c= B )
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 .: ([:NAT,NAT:] \ A) c= B )

hereby :: thesis: ( ( for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s .: ([:NAT,NAT:] \ A) c= B ) implies 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 .: ([:NAT,NAT:] \ A) c= B

hereby :: thesis: verum

let B be Element of cB; :: thesis: ex A being finite Subset of [:NAT,NAT:] st s .: ([:NAT,NAT:] \ A) c= B
consider A being finite Subset of [:NAT,NAT:] such that
A2: s " B = [:NAT,NAT:] \ A by A1, Th49;
take A = A; :: thesis: s .: ([:NAT,NAT:] \ A) c= B
thus s .: ([:NAT,NAT:] \ A) c= B by A2, FUNCT_1:75; :: thesis: verum

end;

end;

assume A3: for B being Element of cB ex A being finite Subset of [:NAT,NAT:] st s .: ([:NAT,NAT:] \ A) c= B ; :: thesis: x in lim_filter (s,(Frechet_Filter [:NAT,NAT:]))
for A being a_neighborhood of x holds [:NAT,NAT:] \ (s " A) is finite

proof

let A be a_neighborhood of x; :: thesis: [:NAT,NAT:] \ (s " A) is finite
A4: A is Element of BOOL2F (NeighborhoodSystem x) by YELLOW19:2;
cB is filter_basis ;
then consider B being Element of cB such that
A5: B c= A by A4;
consider C being finite Subset of [:NAT,NAT:] such that
A6: s .: ([:NAT,NAT:] \ C) c= B by A3;
s .: ([:NAT,NAT:] \ C) c= A by A6, A5;
then A7: s " (s .: ([:NAT,NAT:] \ C)) c= s " A by RELAT_1:143;
dom s = [:NAT,NAT:] by FUNCT_2:def 1;
then [:NAT,NAT:] \ C c= s " (s .: ([:NAT,NAT:] \ C)) by FUNCT_1:76;
then [:NAT,NAT:] \ C c= s " A by A7;
then [:NAT,NAT:] \ (s " A) c= [:NAT,NAT:] \ ([:NAT,NAT:] \ C) by XBOOLE_1:34;
then [:NAT,NAT:] \ (s " A) c= [:NAT,NAT:] /\ C by XBOOLE_1:48;
hence [:NAT,NAT:] \ (s " A) is finite ; :: thesis: verum

end;

hence x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) by Th46; :: thesis: verum