reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th25:
  for T be non empty TopSpace holds T is countably_compact iff for
  F be Subset-Family of T st F is locally_finite & F is with_non-empty_elements
  holds F is finite
proof
  let T be non empty TopSpace;
  thus T is countably_compact implies for F be Subset-Family of T st F is
  locally_finite & F is with_non-empty_elements holds F is finite by Lm2;
  assume for F be Subset-Family of T st F is locally_finite & F is
  with_non-empty_elements holds F is finite;
  then
  for F be Subset-Family of T st F is locally_finite & for A be Subset of
  T st A in F holds card A = 1 holds F is finite by Lm3;
  hence thesis by Lm6;
end;
