let T be non empty TopSpace; :: thesis: for s being Function of , the carrier of T
for x being Point of T
for cB being basis of () holds
( x in lim_filter (s,) iff for B being Element of cB ex A being finite Subset of st s " B = \ A )

let s be Function of , the carrier of T; :: thesis: for x being Point of T
for cB being basis of () holds
( x in lim_filter (s,) iff for B being Element of cB ex A being finite Subset of st s " B = \ A )

let x be Point of T; :: thesis: for cB being basis of () holds
( x in lim_filter (s,) iff for B being Element of cB ex A being finite Subset of st s " B = \ A )

let cB be basis of (); :: thesis: ( x in lim_filter (s,) iff for B being Element of cB ex A being finite Subset of st s " B = \ A )
hereby :: thesis: ( ( for B being Element of cB ex A being finite Subset of st s " B = \ A ) implies x in lim_filter (s,) )
assume A1: x in lim_filter (s,) ; :: thesis: for B being Element of cB ex A being finite Subset of st s " B = \ A
hereby :: thesis: verum
let B be Element of cB; :: thesis: ex A being finite Subset of st s " B = \ A
B is a_neighborhood of x by YELLOW19:2;
hence ex A being finite Subset of st s " B = \ A by ; :: thesis: verum
end;
end;
assume A2: for B being Element of cB ex A being finite Subset of st s " B = \ A ; :: thesis: x in lim_filter (s,)
now :: thesis: for A being a_neighborhood of x holds \ (s " A) is finite
let A be a_neighborhood of x; :: thesis: \ (s " A) is finite
A3: A is Element of BOOL2F 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 st s " B = \ C by A2;
hence [::] \ (s " A) is finite by ; :: thesis: verum
end;
hence x in lim_filter (s,) by Th46; :: thesis: verum