reserve T for TopSpace;
reserve T for non empty TopSpace;
reserve F for Subset-Family of T;

theorem Th56:
  for F being Subset-Family of T holds (for A being Subset of T st
A in F holds A c= Cl Int A) implies union F c= Cl Int(union F) & Cl(union F) =
  Cl Int Cl(union F)
proof
  let F be Subset-Family of T;
A1: Cl Int Cl(union F) c= Cl Cl(union F) by PRE_TOPC:19,TOPS_1:16;
  assume
A2: for A being Subset of T st A in F holds A c= Cl Int A;
A3: now
    let A0 be set;
    assume
A4: A0 in F;
    then reconsider A = A0 as Subset of T;
    Int A c= Int(union F) by A4,TOPS_1:19,ZFMISC_1:74;
    then
A5: Cl Int A c= Cl Int(union F) by PRE_TOPC:19;
    A c= Cl Int A by A2,A4;
    hence A0 c= Cl Int(union F) by A5;
  end;
  hence union F c= Cl Int(union F) by ZFMISC_1:76;
  Int(union F) c= Int Cl(union F) by PRE_TOPC:18,TOPS_1:19;
  then
A6: Cl Int(union F) c= Cl Int Cl(union F) by PRE_TOPC:19;
  union F c= Cl Int(union F) by A3,ZFMISC_1:76;
  then Cl(union F) c= Cl Cl Int(union F) by PRE_TOPC:19;
  then Cl(union F) c= Cl Int Cl(union F) by A6;
  hence Cl(union F) = Cl Int Cl(union F) by A1,XBOOLE_0:def 10;
end;
