reserve T for non empty Abelian
  add-associative right_zeroed right_complementable RLSStruct,
  X,Y,Z,B,C,B1,B2 for Subset of T,
  x,y,p for Point of T;
reserve t,s,r1 for Real;
reserve n for Element of NAT;
reserve X,Y,B1,B2 for Subset of TOP-REAL n;
reserve x,y for Point of TOP-REAL n;

theorem
  for X being non empty Subset of TOP-REAL n holds 0(.)X = {0.TOP-REAL n}
proof
  set T = TOP-REAL n;
  let X be non empty Subset of TOP-REAL n;
  thus 0(.)X c= {0.T}
  proof
    let x be object;
    assume x in 0(.)X;
    then ex a being Point of T st x=0 * a & a in X;
    then x = 0.T by RLVECT_1:10;
    hence thesis by ZFMISC_1:31;
  end;
  let x be object;
  set d = the Element of X;
  reconsider d1=d as Point of T;
  assume x in {0.T};
  then {x} c= {0.T} by ZFMISC_1:31;
  then x = 0.T by ZFMISC_1:18;
  then x = 0 * d1 by RLVECT_1:10;
  hence thesis;
end;
