
theorem Th20:
  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}, x be Vector of X, t,r be
Element of K st w |-- (W,Lin{y}) = [x,r*v] holds t*w |-- (W,Lin{y}) = [t*x, t*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};
  assume 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}, x be Vector of X, t,r be Element of K such
  that
A5: w |-- (W,Lin{y}) = [x,r*v];
  reconsider z = x as Vector of X + Lin{v} by A2,VECTSP_4:10;
  r*y = r*v by A1,VECTSP_4:14;
  then
A6: t*w = t*(z + r*y) by A4,A5,Th6
    .= t*z + t*(r*y) by VECTSP_1:def 14
    .= t*z + t*r*y by VECTSP_1:def 16;
  reconsider u = x as Vector of V by VECTSP_4:10;
A7: t*r*y in Lin{y} by Th3;
A8: t*r*y = t*r*v by A1,VECTSP_4:14;
A9: t*z = t*u by VECTSP_4:14
    .= t*x by VECTSP_4:14;
  then t*z in W by A2;
  hence thesis by A4,A9,A8,A7,A6,Th5;
end;
