let T be non empty TopSpace; :: thesis: for A being Subset of
for x being Point of 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 ; :: thesis: for x being Point of 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 ; :: 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 st
( A in F & x is_a_cluster_point_of F,T )
then consider N being net of such that
A1: N is_eventually_in A and
A2: 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 A2, Th12; :: thesis: verum

end;

given F being proper Filter of BoolePoset ([#] T) such that A3: A in F and
A4: x is_a_cluster_point_of F,T ; :: thesis: x in Cl A
reconsider F' = F as proper Filter of BoolePoset ([#] T) ;
A5: a_filter (a_net F') = F by Th15;
then A6: x is_a_cluster_point_of a_net F' by A4, Th12;
a_net F' is_eventually_in A by A3, A5, Th11;
hence x in Cl A by A6, Th23; :: thesis: verum