reserve T for TopSpace;

theorem
  for F being Subset-Family of T, A being Subset of T holds A in F
  implies Int A c= union(Int F) & meet(Int F) c= Int A
proof
  let F be Subset-Family of T, A be Subset of T;
  assume A in F;
  then
  Int A in {P where P is Subset of T : ex B being Subset of T st P = Int B
  & B in F};
  then
A1: Int A in Int F by Th16;
  hence Int A c= union(Int F) by ZFMISC_1:74;
  thus thesis by A1,SETFAM_1:3;
end;
