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 Th7:
  for Af be Subset of n-VectSp_over F_Real,
      Ar be Subset of TOP-REAL n st Af = Ar
  holds
    Af is linearly-independent
  iff
    Ar is linearly-independent
proof
  set V=n-VectSp_over F_Real;
  let AV be Subset of V;
  set T=TOP-REAL n;
  let AR be Subset of T such that
   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 V by Th1;
    A3: Carrier L1=Carrier L by Th2;
    assume Sum L=0.T;
    then A4: 0.V=Sum L by Lm2
     .=Sum L1 by Th5;
    Carrier L c=AR by RLVECT_2:def 6;
    then L1 is Linear_Combination of AV by A1,A3,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 Th1;
   A6: Carrier L1=Carrier L by Th2;
   Carrier L c=AV by VECTSP_6:def 4;
   then reconsider L1 as Linear_Combination of AR by A1,A6,RLVECT_2:def 6;
   assume Sum L=0.V;
   then 0.T=Sum L by Lm2
    .=Sum L1 by Th5;
   hence Carrier L={} by A5,A6,RLVECT_3:def 1;
  end;
  hence thesis by VECTSP_7:def 1;
end;
