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

theorem Th18:
  for F be add-associative right_zeroed right_complementable
  Abelian associative well-unital right_unital distributive non empty
  doubleLoopStr, V be scalar-distributive vector-distributive
  scalar-associative scalar-unital add-associative right_zeroed
  right_complementable non empty ModuleStr over F, x being Element of F, v
  being Element of V holds x*(-v)=-x*v
proof
  let F be add-associative right_zeroed right_complementable Abelian
associative well-unital right_unital distributive non empty doubleLoopStr, V
  be scalar-distributive vector-distributive scalar-associative scalar-unital
  add-associative right_zeroed right_complementable non empty
  ModuleStr over F, x be Element of F, v be Element of V;
  x*(-v)=x*((-1.F)*v) by Th10
    .=(x*(-1.F))*v by Def15
    .=(-(x*1.F))*v by Th4
    .=(-x)*v;
  hence thesis by Th17;
end;
