let T be TopSpace; :: thesis: for A being Subset of T holds
( A is 1st_class iff A ` is 1st_class )
let A be Subset of T; :: thesis: ( A is 1st_class iff A ` is 1st_class )
A1: ( A ` is 1st_class implies A is 1st_class )
proof
assume A ` is 1st_class ; :: thesis: A is 1st_class
then Int (Cl (A `)) c= Cl (Int (A `)) ;
then Int ((Int A) `) c= Cl (Int (A `)) by TDLAT_3:2;
then (Cl (Int A)) ` c= Cl (Int (A `)) by TDLAT_3:3;
then (Cl (Int A)) ` c= Cl ((Cl A) `) by TDLAT_3:3;
then (Cl (Int A)) ` c= (Int (Cl A)) ` by TDLAT_3:2;
then Int (Cl A) c= Cl (Int A) by SUBSET_1:12;
hence A is 1st_class ; :: thesis: verum
end;
( A is 1st_class implies A ` is 1st_class )
proof
assume A is 1st_class ; :: thesis: A ` is 1st_class
then Int (Cl A) c= Cl (Int A) ;
then (Cl (Int A)) ` c= (Int (Cl A)) ` by SUBSET_1:12;
then Int ((Int A) `) c= (Int (Cl A)) ` by TDLAT_3:3;
then Int ((Int A) `) c= Cl ((Cl A) `) by TDLAT_3:2;
then Int ((Int A) `) c= Cl (Int (A `)) by TDLAT_3:3;
then Int (Cl (A `)) c= Cl (Int (A `)) by TDLAT_3:2;
hence A ` is 1st_class ; :: thesis: verum
end;
hence ( A is 1st_class iff A ` is 1st_class ) by A1; :: thesis: verum