reserve r,s,t,u for Real;

theorem Th59:
  for X being LinearTopSpace, V being a_neighborhood of 0.X ex W
  being a_neighborhood of 0.X st Cl W c= V
proof
  let X be LinearTopSpace, V be a_neighborhood of 0.X;
  set B = the set of all U where U is a_neighborhood of 0.X;
  B c= bool the carrier of X
  proof
    let A be object;
    assume A in B;
    then ex U being a_neighborhood of 0.X st A = U;
    hence thesis;
  end;
  then B is local_base of X by Th44;
  then consider W being a_neighborhood of 0.X such that
  W in B and
A1: Cl W c= V by Th58;
  take W;
  thus thesis by A1;
end;
