
theorem Th26:
  for V being RealUnitarySpace, M,N being non empty Subset of V st
  M c= N holds the carrier of Ort_Comp N c= the carrier of Ort_Comp M
proof
  let V be RealUnitarySpace;
  let M,N be non empty Subset of V;
  assume
A1: M c= N;
    let x be object;
    assume x in the carrier of Ort_Comp N;
    then x in {v where v is VECTOR of V : for w being VECTOR of V st w in N
    holds w, v are_orthogonal} by Def4;
    then
A2: ex v1 being VECTOR of V st x = v1 & for w being VECTOR of V st w in N
    holds w,v1 are_orthogonal;
    then reconsider x as VECTOR of V;
    for y being VECTOR of V st y in M holds y,x are_orthogonal by A1,A2;
    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};
    hence thesis by Def4;
end;
