reserve x,y,y1,y2 for object;

theorem Th4:
  for GF be add-associative right_zeroed right_complementable
  Abelian associative well-unital distributive non empty doubleLoopStr, V be
  Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF holds {0.V} is linearly-closed
proof
  let GF be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over GF;
  thus for v,u being Element of V st v in {0.V} & u in {0.V} holds v + u in {
  0.V}
  proof
    let v,u be Element of V;
    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 by RLVECT_1:4;
    hence thesis by TARSKI:def 1;
  end;
  let a be Element of GF;
  let v be Element of V;
  assume
A1: v in {0.V};
  then v = 0.V by TARSKI:def 1;
  hence thesis by A1,VECTSP_1:14;
end;
