reserve x,y,y1,y2 for object;

theorem Th2:
  for GF be add-associative right_zeroed right_complementable
  Abelian associative well-unital distributive non empty doubleLoopStr, V be
  Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF, V1 be Subset of V st V1 is linearly-closed for v
  being Element of V st v in V1 holds - v in V1
proof
  let GF be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V be Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital non empty
  ModuleStr over GF, V1 be Subset of V;
  assume
A1: V1 is linearly-closed;
  let v be Element of V;
  assume v in V1;
  then (- 1_GF) * v in V1 by A1;
  hence thesis by VECTSP_1:14;
end;
