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

theorem Th84:
  for F being Subset-Family of T holds (for B being Subset of T st
  B in F holds Int(meet F) c= B) & (F = {} or for A being Subset of T st A is
open_condensed holds (for B being Subset of T st B in F holds A c= B) implies A
  c= Int(meet F))
proof
  let F be Subset-Family of T;
  thus for B being Subset of T st B in F holds Int(meet F) c= B
  proof
    let B be Subset of T;
    assume B in F;
    then
A1: meet F c= B by SETFAM_1:3;
    Int(meet F) c= meet F by TOPS_1:16;
    hence thesis by A1;
  end;
  thus F = {} or for A being Subset of T st A is open_condensed holds (for B
  being Subset of T st B in F holds A c= B) implies A c= Int(meet F)
  proof
    assume
A2: F <> {};
    let A be Subset of T;
    assume
A3: A is open_condensed;
    assume for B being Subset of T st B in F holds A c= B;
    then for P be set st P in F holds A c= P;
    then
A4: A c= meet F by A2,SETFAM_1:5;
    A is open by A3,TOPS_1:67;
    then Int A = A by TOPS_1:23;
    hence thesis by A4,TOPS_1:19;
  end;
end;
