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

theorem Th26:
  for T be non empty TopSpace holds T is countably_compact iff 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
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 & for A be Subset of T st A in F holds card A=1 holds F is
  finite
  proof
    assume T is countably_compact;
    then for F be Subset-Family of T st F is locally_finite & F is
    with_non-empty_elements holds F is finite by Th25;
    hence thesis by Lm3;
  end;
  thus thesis by Lm6;
end;
