reserve r,s,t,u for Real;

theorem
  for X being LinearTopSpace, B being circled Subset of X st 0.X in Int
  B holds Int B is circled
proof
  let X be LinearTopSpace, B be circled Subset of X such that
A1: 0.X in Int B;
  let r be Real;
  assume |.r.| <= 1;
  then r*B c= B by Def6;
  then
A2: Int(r*B) c= Int(B) by TOPS_1:19;
  per cases;
  suppose
    r = 0;
    then r*Int(B) = {0.X} by A1,CONVEX1:34;
    hence thesis by A1,ZFMISC_1:31;
  end;
  suppose
    r <> 0;
    hence thesis by A2,Th51;
  end;
end;
