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

theorem Th77:
  for F being Subset-Family of T holds (F is closed implies for B
being Subset of T st B in F holds Cl Int(meet F) c= B) & (F = {} or for A being
  Subset of T st A is closed_condensed holds (for B being Subset of T st B in F
  holds A c= B) implies A c= Cl Int(meet F))
proof
  let F be Subset-Family of T;
  thus F is closed implies for B being Subset of T st B in F holds Cl Int(meet
  F) c= B
  proof
    assume F is closed;
    then meet F is closed by TOPS_2:22;
    then
A1: Cl meet F = meet F by PRE_TOPC:22;
    let B be Subset of T;
A2: Cl Int(meet F) c= Cl meet F by PRE_TOPC:19,TOPS_1:16;
    assume B in F;
    then meet F c= B by SETFAM_1:3;
    hence thesis by A2,A1;
  end;
  thus F = {} or for A being Subset of T st A is closed_condensed holds (for B
  being Subset of T st B in F holds A c= B) implies A c= Cl Int(meet F)
  proof
    assume
A3: F <> {};
    let A be Subset of T;
    assume
A4: A is closed_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 A c= meet F by A3,SETFAM_1:5;
    then
A5: Int A c= Int(meet F) by TOPS_1:19;
    A = Cl Int A by A4,TOPS_1:def 7;
    hence thesis by A5,PRE_TOPC:19;
  end;
end;
