
theorem Th25:
  for V being RealUnitarySpace, M being non empty Subset of V
  holds M c= the carrier of Ort_Comp (Ort_Comp M)
proof
  let V be RealUnitarySpace;
  let M be non empty Subset of V;
    let x be object;
    assume
A1: x in M;
    then reconsider x as VECTOR of V;
    for y being VECTOR of V st y in Ort_Comp M holds x,y are_orthogonal
    proof
      let y be VECTOR of V;
      assume y in Ort_Comp M;
      then y in the carrier of Ort_Comp M;
      then y in {v where v is VECTOR of V : for w being VECTOR of V st w in M
      holds w, v are_orthogonal} by Def4;
      then ex v being VECTOR of V st y = v & for w being VECTOR of V st w in M
      holds w, v are_orthogonal;
      hence thesis by A1;
    end;
    then x in {v where v is VECTOR of V : for w being VECTOR of V st w in
    Ort_Comp M holds w, v are_orthogonal};
    hence thesis by Def3;
end;
