reserve r,s,t,u for Real;

theorem Th27:
  for X being RealLinearSpace, B being non empty circled Subset of
  X holds 0.X in B
proof
  let X be RealLinearSpace;
  let B be non empty circled Subset of X;
  |. 0 .| = 0 by ABSVALUE:2;
  then 0*B c= B by Def6;
  then
A1: {0.X} c= B by CONVEX1:34;
  0.X in {0.X} by TARSKI:def 1;
  hence thesis by A1;
end;
