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;

theorem
  x in (0).V iff x = 0.V
proof
  thus x in (0).V implies x = 0.V
  proof
    assume x in (0).V;
    then x in the carrier of (0).V;
    then x in {0.V} by Def3;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by Th17;
end;
