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
  for W be RealLinearSpace holds
    W is finite-dimensional
      iff
    RLSp2RVSp(W) is finite-dimensional
  proof
    let W be RealLinearSpace;
    hereby
      assume W is finite-dimensional;
      then ex A be finite Subset of W st
      A is Basis of W by RLVECT_5:def 1;
      hence RLSp2RVSp(W) is finite-dimensional by Th80;
    end;
    thus thesis by Th80,RLVECT_5:def 1;
  end;
