reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th21:
  for Af be Subset of RealVectSpace(Seg n),
      Ar be Subset of TOP-REAL n st Af = Ar
  holds Af is linearly-independent iff Ar is linearly-independent
proof
  set V=RealVectSpace(Seg n);
  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 Th17;
      assume Sum L=0.T;
      then
A3: 0.V=Sum L by Lm2 .=Sum L1 by Th20;
      Carrier L c=AR by RLVECT_2:def 6;
      then L1 is Linear_Combination of AV by A1,RLVECT_2:def 6;
      hence Carrier L={} by A2,A3,RLVECT_3:def 1;
    end;
    hence AR is linearly-independent by RLVECT_3:def 1;
  end;
  assume
A4: AR is linearly-independent;
  now let L be Linear_Combination of AV;
    reconsider L1=L as Linear_Combination of T by Th17;
    Carrier L c=AV by RLVECT_2:def 6;
    then reconsider L1 as Linear_Combination of AR by A1,RLVECT_2:def 6;
    assume Sum L=0.V;
    then 0.T=Sum L by Lm2 .=Sum L1 by Th20;
    hence Carrier L={} by A4,RLVECT_3:def 1;
  end;
  hence thesis by RLVECT_3:def 1;
end;
