theorem Th4:
  {0.V} is linearly-closed
proof
  thus for v,u st v in {0.V} & u in {0.V} holds v + u in {0.V}
  proof
    let v,u;
    assume v in {0.V} & u in {0.V};
    then v = 0.V & u = 0.V by TARSKI:def 1;
    then v + u = 0.V;
    hence thesis by TARSKI:def 1;
  end;
  let a,v;
  assume
A1: v in {0.V};
  then v = 0.V by TARSKI:def 1;
  hence thesis by A1;
end;
