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 Th61:
  for X be set holds
    X is Basis of TOP-REAL n
      iff
    X is Basis of REAL-NS n
  proof
    let X be set;
    set V = TOP-REAL n;
    set W = REAL-NS n;

    hereby
      assume X is Basis of V;

      then
      reconsider A = X as Basis of V;
      reconsider B = A as Subset of W by Th4;
      A1: A is linearly-independent
        & Lin A = RLSStruct(# the carrier of V,
                              the ZeroF of V,
                              the addF of V,
                              the Mult of V #) by RLVECT_3:def 3;
      then
      A2: B is linearly-independent by Th28;
      set W0 = (Omega). W;

      A3: W0 = RLSStruct(# the carrier of W,
                           the ZeroF of W,
                           the addF of W,
                           the Mult of W #) by RLSUB_1:def 4;

      A4: Lin B is strict Subspace of W0 by Th49;

      A5: [#]Lin A = the carrier of W by A1,Th4;

      the carrier of (Lin B)
       = [#](Lin B)
      .= the carrier of W0 by A3,A5,Th26;

      then
      Lin B
       = W0 by A4,RLSUB_1:32
      .= RLSStruct(# the carrier of W,
                     the ZeroF of W,
                     the addF of W,
                     the Mult of W #) by RLSUB_1:def 4;
      hence X is Basis of W by A2,RLVECT_3:def 3;
    end;
    assume X is Basis of W; then
    reconsider A = X as Basis of W;
    reconsider B = A as Subset of V by Th4;
    A6: A is linearly-independent
      & Lin A = RLSStruct(# the carrier of W,
                            the ZeroF of W,
                            the addF of W,
                            the Mult of W #) by RLVECT_3:def 3;
    then
    A7: B is linearly-independent by Th28;
    set V0 = (Omega). V;

    A8: V0 = RLSStruct(# the carrier of V,
                         the ZeroF of V,
                         the addF of V,
                         the Mult of V #) by RLSUB_1:def 4;

    A9: Lin B is strict Subspace of V0 by Th49;
    A10: [#]Lin A = the carrier of V by A6,Th4;

    the carrier of (Lin B)
     = [#](Lin B)
    .= the carrier of V0 by A8,A10,Th26; then
    Lin B = V0 by A9,RLSUB_1:32
    .= RLSStruct(# the carrier of V,
                   the ZeroF of V,
                   the addF of V,
                   the Mult of V #) by RLSUB_1:def 4;
    hence
    X is Basis of V by A7,RLVECT_3:def 3;
  end;
