reserve V for RealLinearSpace;

theorem Th13:
  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 for
  w being VECTOR of X + Lin{v} st w in X holds w |-- (W,Lin{y}) = [w,0.V]
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} such that
A1: v = y and
A2: X = W and
A3: not v in X;
A4: X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,A2,A3,Th11;
  let w be VECTOR of X + Lin{v};
  assume w in X;
  then w |-- (W,Lin{y}) = [w,0.(X + Lin{v})] by A2,A4,Th6;
  hence thesis by RLSUB_1:11;
end;
