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 Th28:
  for Ar be Subset of REAL-NS n,
      At be Subset of TOP-REAL n
    st Ar = At
  holds
    Ar is linearly-independent
      iff
    At is linearly-independent
  proof
    let Ar be Subset of REAL-NS n,
        At be Subset of TOP-REAL n;
    assume
    A1: Ar = At;
    hereby
      assume
      A2: Ar is linearly-independent;

      now
        let L be Linear_Combination of At;
        reconsider L1 = L as Linear_Combination of REAL-NS n by Th11;
        A3: Carrier L1 = Carrier L;
        assume Sum L = 0. (TOP-REAL n); then
        A4: 0. (REAL-NS n) = Sum L by Th6
        .= Sum L1 by Th23;
        L1 is Linear_Combination of Ar by A1,A3,RLVECT_2:def 6;

        hence Carrier L = {} by A2,A4,RLVECT_3:def 1;
      end;
      hence At is linearly-independent by RLVECT_3:def 1;
    end;

    assume
    A5: At is linearly-independent;
    now
      let L be Linear_Combination of Ar;
      reconsider L1 = L as Linear_Combination of TOP-REAL n by Th11;
      A6: Carrier L1 = Carrier L;
      reconsider L1 = L as Linear_Combination of At by A1,A6,RLVECT_2:def 6;

      assume Sum L = 0.(REAL-NS n);
      then 0. (TOP-REAL n)
       = Sum L by Th6
      .= Sum L1 by Th23;
      hence Carrier L = {} by A5,RLVECT_3:def 1;
    end;
    hence Ar is linearly-independent by RLVECT_3:def 1;
  end;
