reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace ex P being local_base of X st P is
  circled-membered
proof
  let X be LinearTopSpace;
  defpred P[Subset of X] means $1 is circled;
  consider P being Subset-Family of X such that
A1: for V being Subset of X holds V in P iff P[V] from SUBSET_1:sch 3;
  reconsider P as Subset-Family of X;
  take P;
  thus P is local_base of X
  proof
    let V be a_neighborhood of 0.X;
    consider W being a_neighborhood of 0.X such that
A2: W is circled and
A3: W c= V by Th64;
    take W;
    thus W in P by A1,A2;
    thus thesis by A3;
  end;
  let V be Subset of X;
  assume V in P;
  hence thesis by A1;
end;
