
theorem
  for V being RealNormSpace,
      M be Subspace of DualSp V,
      v be VECTOR of V,
      m being VECTOR of DualSp V
   st v in Ort_Comp M & m in M
     holds v .|. m = 0
proof
  let V be RealNormSpace,
      M be Subspace of DualSp V,
      v be VECTOR of V,
      m be VECTOR of DualSp V;
  assume A1: v in Ort_Comp M & m in M;
  the carrier of Ort_Comp M =
  { v where v is VECTOR of V :
  for w being VECTOR of DualSp V st w in M holds
  v,w are_orthogonal } by Def6; then
  v in { v where v is VECTOR of V :
  for w being VECTOR of DualSp V st w in M holds
  v,w are_orthogonal } by A1,STRUCT_0:def 5; then
  ex v0 be VECTOR of V st v = v0 &
  for w being VECTOR of DualSp V st w in M holds
  v0,w are_orthogonal; then
  v,m are_orthogonal by A1;
  hence v .|. m = 0;
end;
