reserve V for non empty RLSStruct;
reserve x,y,y1 for set;
reserve v for VECTOR of V;
reserve a,b for Real;

theorem Th25:
  for V being add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v being Element of V holds
  a * (- v) = - (a * v)
proof
  let V be add-associative right_zeroed right_complementable Abelian
  scalar-distributive scalar-associative scalar-unital vector-distributive
  non empty RLSStruct,
  v be Element of V;
  thus a * (- v) = (- (1 * a)) * v by Th24
    .= ((- 1) * a) * v
    .= (- 1) * (a * v) by Def7
    .= - (a * v) by Th16;
end;
