reserve r,s,t,u for Real;

theorem Th64:
  for X being LinearTopSpace, U being a_neighborhood of 0.X ex W
  being a_neighborhood of 0.X st W is circled & W c= U
proof
  let X be LinearTopSpace, U be a_neighborhood of 0.X;
  0.X = 0*0.X;
  then consider r being positive Real,
    V being a_neighborhood of 0.X such that
A1: for s being Real st |.s-0 .| < r holds s*V c= U by Th32;
  set F = {a*V where a is Real: |.a.| < r};
  F c= bool the carrier of X
  proof
    let A be object;
    assume A in F;
    then ex a being Real st A = a*V & |.a.| < r;
    hence thesis;
  end;
  then reconsider F as Subset-Family of X;
  take union F;
  now
    let s be Real;
    assume |.s.| < r;
    then |.s-0 .| < r;
    hence s*V c= U by A1;
  end;
  hence thesis by Lm20;
end;
