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 n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) 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 n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) 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 n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) 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 n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B )

hereby :: thesis: ( ( for B being Element of cB ex n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) 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 n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B

now :: thesis: for B being Element of cB ex n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B

let B be Element of cB; :: thesis: ex n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B
consider A being finite Subset of [:NAT,NAT:] such that
A2: s .: ([:NAT,NAT:] \ A) c= B by A1, Th51;
consider n, m being Nat such that
A3: A c= [:(Segm n),(Segm m):] by Th16;
[:NAT,NAT:] \ [:(Segm n),(Segm m):] c= [:NAT,NAT:] \ A by A3, XBOOLE_1:34;
then s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= s .: ([:NAT,NAT:] \ A) by RELAT_1:123;
then s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B by A2;
hence ex n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B ; :: thesis: verum

end;

hence for B being Element of cB ex n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B ; :: thesis: verum

end;

assume A4: for B being Element of cB ex n, m being Nat st s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B ; :: thesis: x in lim_filter (s,(Frechet_Filter [:NAT,NAT:]))

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

let B be Element of cB; :: thesis: ex A being finite Subset of [:NAT,NAT:] st s .: ([:NAT,NAT:] \ A) c= B
consider n, m being Nat such that
A5: s .: ([:NAT,NAT:] \ [:(Segm n),(Segm m):]) c= B by A4;
reconsider A = [:(Segm n),(Segm m):] as finite Subset of [:NAT,NAT:] ;
thus ex A being finite Subset of [:NAT,NAT:] st s .: ([:NAT,NAT:] \ A) c= B by A5; :: thesis: verum

end;

hence x in lim_filter (s,(Frechet_Filter [:NAT,NAT:])) by Th51; :: thesis: verum