
theorem Th16:
  for V being RealNormSpace,
      M being Subspace of V,
      v being VECTOR of DualSp V,
      m being VECTOR of V
    st v in Ort_Comp M & m in M holds m .|. v = 0
proof
  let V be RealNormSpace,
      M be Subspace of V,
      v be VECTOR of DualSp V,
      m be VECTOR of V;
   assume A1: v in Ort_Comp M & m in M;
   the carrier of Ort_Comp M =
   { v where v is VECTOR of DualSp V :
   for w being VECTOR of V st w in M holds
   w,v are_orthogonal } by Def5; then
   v in { v where v is VECTOR of DualSp V :
   for w being VECTOR of V st w in M holds
   w,v are_orthogonal } by A1,STRUCT_0:def 5; then
   ex v0 be VECTOR of DualSp V st v = v0 &
   for w being VECTOR of V st w in M holds
   w,v0 are_orthogonal; then
   m,v are_orthogonal by A1;
   hence m .|. v = 0;
 end;
