reserve T for TopSpace;

theorem Th28:
  for F being Subset-Family of T holds Int(meet F) c= meet(Int F)
proof
  let F be Subset-Family of T;
A1: for A being set st A in Int F holds Int(meet F) c= A
  proof
    let A be set;
    assume
A2: A in Int F;
    then reconsider A0 = A as Subset of T;
    A0 in {B where B is Subset of T : ex C being Subset of T st B = Int C
    & C in F} by A2,Th16;
    then
    ex P being Subset of T st P = A0 & ex C being Subset of T st P = Int
    C & C in F;
    hence thesis by SETFAM_1:3,TOPS_1:19;
  end;
  now
    per cases;
    suppose
   Int F = {};
      then meet F = {}T by Th18,SETFAM_1:1;
      hence thesis;
    end;
    suppose
      Int F <> {};
      hence thesis by A1,SETFAM_1:5;
    end;
  end;
  hence thesis;
end;
