reserve V for RealLinearSpace;

theorem Th3:
  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 = v1 + v2
proof
  let W1,W2 be Subspace of V such that
A1: V is_the_direct_sum_of W1,W2;
  let v,v1,v2 be VECTOR of V;
  assume v |-- (W1,W2) = [v1,v2];
  then (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2;
  hence thesis by A1,RLSUB_2:def 6;
end;
