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

theorem Th17:
  for F be add-associative right_zeroed right_complementable
  Abelian associative well-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,w being Element of V holds -x*v=(-x)
  *v & w-x*v=w+(-x)*v
proof
  let F be add-associative right_zeroed right_complementable Abelian
  associative well-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,w be Element of V;
A1: -x*v=(-1.F)*(x*v) by Th10
    .=((-1.F)*x)*v by Def15
    .=(-(1.F*x))*v by Th5;
  hence -x*v=(-x)*v;
  thus thesis by A1;
end;
