reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;

theorem Th41:
  A is affinely-independent iff for v st v in A holds
                                      -v + A\{0.V} is linearly-independent
 proof
  hereby assume A is affinely-independent;
   then for L be Linear_Combination of A st Sum L=0.V & sum L=0 holds Carrier L
={} by Lm5;
   hence for v st v in A holds(-v+A)\{0.V} is linearly-independent by Lm6;
  end;
  assume A1: for v st v in A holds(-v+A)\{0.V} is linearly-independent;
  assume A is non empty;
  then consider v be object such that
   A2: v in A;
  reconsider v as Element of V by A2;
  take v;
  thus thesis by A1,A2;
 end;
