
theorem Lm814:
  for X be RealUnitarySpace, M be Subspace of X,
      x be Point of X
    st x in (the carrier of M) /\ the carrier of (Ort_Comp M)
   holds x = 0.X
proof
  let X be RealUnitarySpace, M be Subspace of X,
      x be Point of X;
  assume x in (the carrier of M) /\ the carrier of (Ort_Comp M); then
A1: x in M & x in Ort_Comp M by XBOOLE_0:def 4; then
  x in {v where v is VECTOR of X : for w being VECTOR of X st
         w in M holds w, v are_orthogonal} by RUSUB_5:def 3; then
  consider v be VECTOR of X such that
A2: x = v &
    for w being VECTOR of X st w in M holds w, v are_orthogonal;
  x, x are_orthogonal by A1,A2;
  hence thesis by BHSP_1:def 2;
end;
