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 Th53: ::: see also RLAFFIN3:4, not really used here
  TOP-REAL n is finite-dimensional
   &
  dim TOP-REAL n = n
  proof
    set V = n -VectSp_over F_Real;
    set W = TOP-REAL n;
    A1: dim V = n by MATRIX13:112;
    consider A be finite Subset of V such that
    A2: A is Basis of V by MATRLIN:def 1;
    A3: card A = n by A1,A2,VECTSP_9:def 1;
    reconsider B = A as finite Subset of W by Lm1;
    thus W is finite-dimensional;
    B is Basis of W by A2,Th50;
    hence dim W = n by A3,RLVECT_5:def 2;
  end;
