reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, U being a_neighborhood of 0.X st U
  is convex ex W being a_neighborhood of 0.X st W is circled convex & W c= U
proof
  let X be LinearTopSpace, U be a_neighborhood of 0.X such that
A1: U is convex;
  set V = U /\ -U;
  -U is a_neighborhood of 0.X by Th55;
  then reconsider V as a_neighborhood of 0.X by CONNSP_2:2;
  take V;
A2: -U is convex by A1,CONVEX1:1;
  thus V is circled
  proof
    0.X in V by CONNSP_2:4;
    then
A3: 0.X in -U by XBOOLE_0:def 4;
A4: 0.X in U by CONNSP_2:4;
    let r be Real such that
A5: |.r.| <= 1;
    let u be object;
    assume u in r*V;
    then consider v being Point of X such that
A6: u = r*v and
A7: v in V;
A8: v in -U by A7,XBOOLE_0:def 4;
A9: v in U by A7,XBOOLE_0:def 4;
    per cases;
    suppose
A10:  r < 0;
      then
A11:  -r <= 1 by A5,ABSVALUE:def 1;
      then (-r)*v + (1--r)*0.X in -U by A2,A8,A3,A10;
      then (-r)*v + 0.X in -U;
      then (-r)*v in -U;
      then consider u9 being Point of X such that
A12:  (-r)*v = (-1)*u9 and
A13:  u9 in U;
      (-r)*v + (1--r)*0.X in U by A1,A9,A4,A10,A11;
      then (-r)*v + 0.X in U;
      then (-r)*v in U;
      then (-1)*(((-1)*r)*v) in (-1)*U;
      then
A14:  ((-1)*((-1)*r))*v in (-1)*U by RLVECT_1:def 7;
      u9 = ((-1)*(-1))*u9 by RLVECT_1:def 8
        .= (-1)*((-1)*u9) by RLVECT_1:def 7
        .= ((-r)*(-1))*v by A12,RLVECT_1:def 7
        .= r*v;
      hence thesis by A6,A13,A14,XBOOLE_0:def 4;
    end;
    suppose
A15:  r >= 0;
A16:  r <= 1 by A5,ABSVALUE:def 1;
      then r*v + (1-r)*0.X in -U by A2,A8,A3,A15;
      then r*v + 0.X in -U;
      then
A17:  r*v in -U;
      r*v + (1-r)*0.X in U by A1,A9,A4,A15,A16;
      then r*v + 0.X in U;
      then r*v in U;
      hence thesis by A6,A17,XBOOLE_0:def 4;
    end;
  end;
  thus V is convex by A1,A2;
  thus thesis by XBOOLE_1:17;
end;
