reserve V for RealLinearSpace;

theorem Th5:
  for W1,W2 being Subspace of V st V is_the_direct_sum_of W1,W2
for v,v1,v2 being VECTOR of V st v |-- (W1,W2) = [v1,v2] holds v |-- (W2,W1) =
  [v2,v1]
proof
  let W1,W2 be Subspace of V;
  assume
A1: V is_the_direct_sum_of W1,W2;
  let v,v1,v2 be VECTOR of V;
  assume
A2: v |-- (W1,W2) = [v1,v2];
  then
A3: (v |-- (W1,W2))`1 = v1;
  then
A4: v1 in W1 by A1,RLSUB_2:def 6;
A5: (v |-- (W1,W2))`2 = v2 by A2;
  then
A6: v2 in W2 by A1,RLSUB_2:def 6;
  v = v2 + v1 by A1,A3,A5,RLSUB_2:def 6;
  hence thesis by A1,A4,A6,Th2,RLSUB_2:38;
end;
