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;
reserve I for affinely-independent Subset of V;

theorem Th46:
  0.V in A implies
    (A is affinely-independent iff A \ {0.V} is linearly-independent)
 proof
  assume A1: 0.V in A;
  A2: -0.V+A=0.V+A
   .=A by Th6;
  hence A is affinely-independent implies A\{0.V} is linearly-independent by A1
,Th41;
  assume A\{0.V} is linearly-independent;
  hence thesis by A1,A2;
 end;
