theorem
  for v be Element of R holds conv {v} = {v}
  proof
    let v be Element of R;
    {v} is convex
    proof
      let u1,u2 be VECTOR of R,r;
      assume that
      0<r and
      r<1;
      assume that
      A1: u1 in {v} and
      A2: u2 in {v};
      u1=v & u2=v by A1,A2,TARSKI:def 1;
      then r*u1+(1-r)*u2=(r+(1-r))*u1 by RLVECT_1:def 6
      .=u1 by RLVECT_1:def 8;
      hence thesis by A1;
    end;
    then conv{v}c={v} by CONVEX1:30;
    hence thesis by ZFMISC_1:33;
  end;
