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 Th81:
  for W be RealLinearSpace
   st W is finite-dimensional
  holds
    RLSp2RVSp(W) is finite-dimensional
      &
    dim RLSp2RVSp(W) = dim W
  proof
    let W be RealLinearSpace;
    assume
    A1: W is finite-dimensional;
    then consider A be finite Subset of W such that
    A2: A is Basis of W by RLVECT_5:def 1;

    reconsider B = A as finite Subset of RLSp2RVSp(W);
    A3: B is Basis of RLSp2RVSp(W) by A2,Th80;
    thus RLSp2RVSp (W) is finite-dimensional by A2,Th80;
    hence dim RLSp2RVSp (W)
     = card B by A3,VECTSP_9:def 1
    .= dim W by A1,A2,RLVECT_5:def 2;
  end;
