
theorem Th27:
  for V being RealUnitarySpace, W being Subspace of V, M being non
  empty Subset of V st M = the carrier of W holds Ort_Comp M = Ort_Comp W
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  let M be non empty Subset of V;
  assume
A1: M = the carrier of W;
  for x being object st x in the carrier of Ort_Comp W holds x in the
  carrier of Ort_Comp M
  proof
    let x be object;
    assume x in the carrier of Ort_Comp W;
    then x in {v where v is VECTOR of V : for w being VECTOR of V st w in W
    holds w, v are_orthogonal} by Def3;
    then consider v being VECTOR of V such that
A2: x = v and
A3: for w being VECTOR of V st w in W holds w,v are_orthogonal;
    for w being VECTOR of V st w in M holds w,v are_orthogonal
    by A1,STRUCT_0:def 5,A3;
    then x in {v1 where v1 is VECTOR of V : for w being VECTOR of V st w in M
    holds w, v1 are_orthogonal} by A2;
    hence thesis by Def4;
  end;
  then
A4: the carrier of Ort_Comp W c= the carrier of Ort_Comp M;
  for x being object st x in the carrier of Ort_Comp M
    holds x in the carrier of Ort_Comp W
  proof
    let x be object;
    assume x in the carrier of Ort_Comp M;
    then x 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 consider v being VECTOR of V such that
A5: x = v and
A6: for w being VECTOR of V st w in M holds w,v are_orthogonal;
    for w being VECTOR of V st w in W holds w,v are_orthogonal
    by A1,A6;
    then x in {v1 where v1 is VECTOR of V : for w being VECTOR of V st w in W
    holds w, v1 are_orthogonal} by A5;
    hence thesis by Def3;
  end;
  then the carrier of Ort_Comp M c= the carrier of Ort_Comp W;
  then the carrier of Ort_Comp W = the carrier of Ort_Comp M by A4;
  hence thesis by RUSUB_1:24;
end;
