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 Th42:
  A is affinely-independent iff
      for L be Linear_Combination of A st Sum L = 0.V & sum L = 0
        holds Carrier L = {}
 proof
  thus A is affinely-independent implies for L be Linear_Combination of A st
  Sum L=0.V & sum L=0 holds Carrier L={} by Lm5;
  assume for L be Linear_Combination of A st Sum L=0.V & sum L=0 holds Carrier
L={};
  then for p be VECTOR of V st p in A holds(-p+A)\{0.V} is linearly-independent
by Lm6;
  hence thesis by Th41;
 end;
