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

theorem
  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,w being Element of V holds x*(v-w)=x
  *v-x*w
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,w be Element of V;
  x*(v-w)=x*v+x*(-w) by Def13
    .=x*v+(-x*w) by Th18;
  hence thesis;
end;
