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 Th41:
  for X be set
  holds
    X is affinely-independent Subset of REAL-NS n
      iff
    X is affinely-independent Subset of TOP-REAL n
  proof
    let X be set;

    hereby
      assume X is affinely-independent Subset of REAL-NS n;
      then
      reconsider Ar = X as affinely-independent Subset of REAL-NS n;
      reconsider At = Ar as Subset of TOP-REAL n by Th4;

      per cases by RLAFFIN1:def 4;
      suppose
        Ar is empty; then
        At is empty;
        hence X is affinely-independent Subset of TOP-REAL n;
      end;

      suppose
        ex v be VECTOR of REAL-NS n
        st v in Ar & ((- v) + Ar) \ {(0. (REAL-NS n))}
          is linearly-independent; then
        consider v be VECTOR of REAL-NS n such that
        A1: v in Ar & ((- v) + Ar) \ {(0. (REAL-NS n))}
          is linearly-independent;

        reconsider w=v as VECTOR of TOP-REAL n by Th4;
        A2 : 0. (REAL-NS n) = 0. (TOP-REAL n) by Th6;

        ((- v) + Ar) \ {(0. (REAL-NS n))}
          = ((- w) + At) \ {(0. (TOP-REAL n))} by A2,Th9,Th39; then
        ((- w) + At) \ {(0. (TOP-REAL n))}
          is linearly-independent by A1,Th28;

        hence X is affinely-independent Subset of TOP-REAL n
          by A1,RLAFFIN1:def 4;
      end;
    end;

    assume X is affinely-independent Subset of TOP-REAL n;

    then
    reconsider At = X as affinely-independent Subset of TOP-REAL n;
    reconsider Ar=At as Subset of REAL-NS n by Th4;

    per cases by RLAFFIN1:def 4;
    suppose
      At is empty; then
      Ar is empty;
      hence X is affinely-independent Subset of REAL-NS n;
    end;
    suppose
      ex v be VECTOR of TOP-REAL n
      st v in At & ((- v) + At) \ {(0. (TOP-REAL n))}
        is linearly-independent; then
      consider v be VECTOR of TOP-REAL n such that
      A3: v in At & ((- v) + At) \ {(0. (TOP-REAL n))}
        is linearly-independent;

      reconsider w=v as VECTOR of REAL-NS n by Th4;
      A4: 0. (REAL-NS n) = 0. (TOP-REAL n) by Th6;
        ((- w) + Ar) \ {(0. (REAL-NS n))}
      = ((- v) + At) \ {(0. (TOP-REAL n))} by A4,Th9,Th39;

      then
      ((- w) + Ar) \ {(0. (REAL-NS n))}
        is linearly-independent by A3,Th28;
      hence
      X is affinely-independent Subset of REAL-NS n by A3,RLAFFIN1:def 4;
    end;
  end;
