let T be non empty TopSpace; :: thesis: for A being Subset of T
for x being Point of T holds
( x in Cl A iff ex F being proper Filter of (BoolePoset ([#] T)) st
( A in F & x is_a_cluster_point_of F,T ) )
let A be Subset of T; :: thesis: for x being Point of T holds
( x in Cl A iff ex F being proper Filter of (BoolePoset ([#] T)) st
( A in F & x is_a_cluster_point_of F,T ) )
let x be Point of T; :: thesis: ( x in Cl A iff ex F being proper Filter of (BoolePoset ([#] T)) st
( A in F & x is_a_cluster_point_of F,T ) )

hereby :: thesis: ( ex F being proper Filter of (BoolePoset ([#] T)) st
( A in F & x is_a_cluster_point_of F,T ) implies x in Cl A )

assume x in Cl A ; :: thesis: ex F being Subset of (BoolePoset ([#] T)) st
( A in F & x is_a_cluster_point_of F,T )
then consider N being net of T such that
A1: ( N is_eventually_in A & x is_a_cluster_point_of N ) by Th23;
set F = a_filter N;
take F = a_filter N; :: thesis: ( A in F & x is_a_cluster_point_of F,T )
thus A in F by A1; :: thesis: x is_a_cluster_point_of F,T
thus x is_a_cluster_point_of F,T by A1, Th12; :: thesis: verum

end;

given F being proper Filter of (BoolePoset ([#] T)) such that A2: ( A in F & x is_a_cluster_point_of F,T ) ; :: thesis: x in Cl A
reconsider F' = F as proper Filter of (BoolePoset ([#] T)) ;
a_filter (a_net F') = F by Th15;
then ( a_net F' is_eventually_in A & x is_a_cluster_point_of a_net F' ) by A2, Th11, Th12;
hence x in Cl A by Th23; :: thesis: verum