reserve V for RealLinearSpace;

theorem Th15:
  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} ex x being VECTOR of X, r being Real
   st w |-- (W,
  Lin{y}) = [x,r*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;
  let w be VECTOR of X + Lin{v};
  consider v1,v2 being VECTOR of X + Lin{v} such that
A4: w |-- (W,Lin{y}) = [v1,v2] by Th14;
A5: X + Lin{v} is_the_direct_sum_of W,Lin{y} by A1,A2,A3,Th11;
  then v1 in W by A4,Th4;
  then reconsider x = v1 as VECTOR of X by A2;
  v2 in Lin{y} by A5,A4,Th4;
  then consider r being Real such that
A6: v2 = r * y by RLVECT_4:8;
  take x,r;
  thus thesis by A1,A4,A6,RLSUB_1:14;
end;
