reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th17:
  for X be finite-dimensional RealLinearSpace,
      b be OrdBasis of RLSp2RVSp(X),
      y be Element of RLSp2RVSp(X)
  holds y |-- b is Element of REAL(dim X)
  proof
    let X be finite-dimensional RealLinearSpace,
        b be OrdBasis of RLSp2RVSp(X),
        y be Element of RLSp2RVSp(X);
    set z = y |-- b;
    len z = len b by MATRLIN:def 7
    .= dim(RLSp2RVSp(X)) by MATRLIN2:21
    .= dim X by REAL_NS2:81;
    hence thesis by FINSEQ_2:92;
  end;
