reserve F for Field,
  x for Element of F,
  V for VectSp of F,
  v for Element of V;

theorem Th10:
  for F being add-associative right_zeroed right_complementable
    Abelian associative well-unital distributive non empty doubleLoopStr,
      x being Element of F
  for V being add-associative right_zeroed right_complementable
    scalar-distributive vector-distributive scalar-associative scalar-unital
        non empty ModuleStr over F,
      v being Element of V holds
   (0.F)*v = 0.V & (-1.F)*v = -v & x*(0.V) = 0.V
proof
  let F be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr;
  let x be Element of F;
  let V be add-associative right_zeroed right_complementable
  scalar-distributive
   vector-distributive scalar-associative scalar-unital non
  empty ModuleStr over F, v be Element of V;
  v+(0.F)*v = (1.F)*v + (0.F)*v by Def16
    .= ((1.F)+(0.F))*v by Def14
    .= (1.F)*v by RLVECT_1:4
    .= v by Def16
    .= v+0.V by RLVECT_1:4;
  hence
A1: (0.F)*v = 0.V by RLVECT_1:8;
  (-(1.F))*v+v = (-(1.F))*v + (1.F)*v by Def16
    .= ((1.F)+(-(1.F)))*v by Def14
    .= 0.V by A1,RLVECT_1:def 10;
  then (-(1.F))*v + (v+(-v)) = 0.V + -v by RLVECT_1:def 3;
  then 0.V + -v = (-(1.F))*v + 0.V by RLVECT_1:5
    .= (-(1.F))*v by RLVECT_1:4;
  hence (-1.F)*v = -v by RLVECT_1:4;
  x*(0.V) = (x*(0.F))*v by A1,Def15
    .= 0.V by A1;
  hence thesis;
end;
