
theorem Th19:
  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 w1,w2 be Vector of X + Lin{v}, x1,x2 be Vector of X,
r1,r2 be Element of K st w1 |-- (W,Lin{y}) = [x1,r1*v] & w2 |-- (W,Lin{y}) = [
  x2,r2*v] holds w1 + w2 |-- (W,Lin{y}) = [x1 + x2, (r1+r2)*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 w1,w2 be Vector of X + Lin{v}, x1,x2 be Vector of X, r1,r2 be Element of
  K such that
A5: w1 |-- (W,Lin{y}) = [x1,r1*v] and
A6: w2 |-- (W,Lin{y}) = [x2,r2*v];
  reconsider y1 = x1, y2 = x2 as Vector of X + Lin{v} by A2,VECTSP_4:10;
A7: r2*v = r2*y by A1,VECTSP_4:14;
  then
A8: y2 in W by A4,A6,Th7;
  (r1+r2)*v = (r1+r2)*y by A1,VECTSP_4:14;
  then
A9: (r1+r2)*v in Lin{y} by Th3;
  reconsider z1 = x1, z2 = x2 as Vector of V by VECTSP_4:10;
A10: y1 + y2 = z1 + z2 by VECTSP_4:13
    .= x1 + x2 by VECTSP_4:13;
A11: r1*v = r1*y by A1,VECTSP_4:14;
  then y1 in W by A4,A5,Th7;
  then
A12: y1 + y2 in W by A8,VECTSP_4:20;
A13: w2 = y2 + r2*y by A4,A6,A7,Th6;
  w1 = y1 + r1*y by A4,A5,A11,Th6;
  then
A14: w1 + w2 = y1 + r1*y + y2 + r2*y by A13,RLVECT_1:def 3
    .= y1 + y2 + r1*y + r2*y by RLVECT_1:def 3
    .= y1 + y2 + (r1*y + r2*y) by RLVECT_1:def 3
    .= y1 + y2 + (r1+r2)*y by VECTSP_1:def 15;
  (r1+r2)*y = (r1+r2)*v by A1,VECTSP_4:14;
  hence thesis by A4,A12,A9,A14,A10,Th5;
end;
