
theorem Th16:
  for K be Field, V be VectSp of K, v be Vector of V, X be
Subspace of V for y be Vector of X + Lin{v}, W be Subspace of X + Lin{v} st v =
y & X = W & not v in X for w be Vector of X + Lin{v} st w in X holds w |-- (W,
  Lin{y}) = [w,0.V]
proof
  let K be Field, V be VectSp of K, 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,Th14;
  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,Th9;
  hence thesis by VECTSP_4:11;
end;
