
theorem Th18:
  for K be Field, V be VectSp of K for 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 for w be Vector of X + Lin{v} ex x be Vector of X, r be
  Element of K st w |-- (W,Lin{y}) = [x,r*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;
  let w be Vector of X + Lin{v};
  consider v1,v2 be Vector of X + Lin{v} such that
A4: w |-- (W,Lin{y}) = [v1,v2] by Th17;
A5: X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,A2,A3,Th14;
  then v1 in W by A4,Th7;
  then reconsider x = v1 as Vector of X by A2;
  v2 in Lin{y} by A5,A4,Th7;
  then consider r be Element of K such that
A6: v2 = r * y by Th3;
  take x,r;
  thus thesis by A1,A4,A6,VECTSP_4:14;
end;
