let T be TopSpace; :: thesis: for A being Subset of T st A is 1st_class holds
( Int (Cl A) = Int (Cl (Int A)) & Cl (Int A) = Cl (Int (Cl A)) )
let A be Subset of T; :: thesis: ( A is 1st_class implies ( Int (Cl A) = Int (Cl (Int A)) & Cl (Int A) = Cl (Int (Cl A)) ) )
assume A is 1st_class ; :: thesis: ( Int (Cl A) = Int (Cl (Int A)) & Cl (Int A) = Cl (Int (Cl A)) )
then A1: Int (Cl A) c= Cl (Int A) by Def6;
then A2: Int (Int (Cl A)) c= Int (Cl (Int A)) by TOPS_1:48;
Cl (Int A) c= Cl A by PRE_TOPC:49, TOPS_1:44;
then A3: Int (Cl (Int A)) c= Int (Cl A) by TOPS_1:48;
A4: Cl (Int (Cl A)) c= Cl (Cl (Int A)) by A1, PRE_TOPC:49;
Int A c= Int (Cl A) by PRE_TOPC:48, TOPS_1:48;
then Cl (Int A) c= Cl (Int (Cl A)) by PRE_TOPC:49;
hence ( Int (Cl A) = Int (Cl (Int A)) & Cl (Int A) = Cl (Int (Cl A)) ) by A2, A3, A4, XBOOLE_0:def 10; :: thesis: verum