
theorem Th15:
  for 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} st v = y
  & X = W & not v in X holds y |-- (W,Lin{y}) = [0.W,y]
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};
  assume v = y & X = W & not v in X;
  then y in {y} & X + Lin{v} is_the_direct_sum_of W,Lin{y} by Th14,TARSKI:def 1
;
  then y |-- (W,Lin{y}) = [0.(X + Lin{v}),y] by Th10,VECTSP_7:8;
  hence thesis by VECTSP_4:11;
end;
