reserve x,y,y1,y2 for object;
reserve GF for add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr,
  V,X,Y for Abelian add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital non
  empty ModuleStr over GF;
reserve a for Element of GF;
reserve u,u1,u2,v,v1,v2 for Element of V;
reserve W,W1,W2 for Subspace of V;
reserve V1 for Subset of V;
reserve w,w1,w2 for Element of W;
reserve B,C for Coset of W;

theorem
  for GF being Field, V being VectSp of GF, v being Element of V, W
  being Subspace of V holds v in W iff - v + W = the carrier of W
proof
  let GF be Field, V be VectSp of GF, v be Element of V, W be Subspace of V;
  - 1_GF <> 0.GF by VECTSP_2:3;
  then v in W iff ((- 1_GF) * v) + W = the carrier of W by Th50,Th51;
  hence thesis by VECTSP_1:14;
end;
