reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;

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;
