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 Th62:
  REAL-NS n is finite-dimensional
    &
  dim (REAL-NS n) = n
  proof
    set V = TOP-REAL n;
    set W = REAL-NS n;
    A1: dim V = n by Th53;
    consider A be finite Subset of V such that
    A3: A is Basis of V by RLVECT_5:def 1;

    A4: card A = n by A1,A3,RLVECT_5:def 2;
    reconsider B = A as finite Subset of W by Th4;
    A5: B is Basis of W by A3,Th61;
    hence W is finite-dimensional by RLVECT_5:def 1;
    hence dim W = n by A4,A5,RLVECT_5:def 2;
  end;
