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 Th78:
  for T be RealLinearSpace,
     Af be Subset of RLSp2RVSp(T),
     Ar be Subset of T
    st Af = Ar
  holds
    Af is linearly-independent
      iff
    Ar is linearly-independent
  proof
    let T be RealLinearSpace;
    let AV be Subset of RLSp2RVSp(T);
    let AR be Subset of T;
    assume
    A1: AV = AR;
    hereby
      assume
      A2: AV is linearly-independent;
      now
        let L be Linear_Combination of AR;
        reconsider L1 = L as Linear_Combination of RLSp2RVSp(T) by Th72;
        A3: Carrier L1 = Carrier L by Th73;
        assume Sum L = 0.T; then
        A4: 0. (RLSp2RVSp(T)) = Sum L1 by Th76;
        L1 is Linear_Combination of AV
          by A1,A3,RLVECT_2:def 6,VECTSP_6:def 4;
        hence Carrier L = {} by A2,A3,A4,VECTSP_7:def 1;
      end;
      hence AR is linearly-independent by RLVECT_3:def 1;
    end;
    assume
    A5: AR is linearly-independent;
    now
      let L be Linear_Combination of AV;
      reconsider L1 = L as Linear_Combination of T by Th72;
      A6: Carrier L1 = Carrier L by Th73;
      reconsider L1 as Linear_Combination of AR
        by VECTSP_6:def 4,A1, A6, RLVECT_2:def 6;
      assume Sum L = 0. (RLSp2RVSp(T));
      then 0. T = Sum L1 by Th76;
      hence Carrier L = {} by A5, A6, RLVECT_3:def 1;
    end;
    hence AV is linearly-independent by VECTSP_7:def 1;
  end;
