reserve V for RealLinearSpace;

theorem Th12:
  for v being VECTOR of V, X being Subspace of V, y being VECTOR
  of X + Lin{v}, W being Subspace of X + Lin{v} st v = y & X = W & not v in X
  holds y |-- (W,Lin{y}) = [0.W,y]
proof
  let v be VECTOR of V, X be Subspace of V, y be VECTOR of X + Lin{v}, W be
  Subspace of X + Lin{v};
  assume v = y & X = W & not v in X;
  then X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th11;
  then y |-- (W,Lin{y}) = [0.(X + Lin{v}),y] by Th7,RLVECT_4:9;
  hence thesis by RLSUB_1:11;
end;
