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 Th43:
  A is affinely-independent & B c= A implies B is affinely-independent
 proof
  assume that
   A1: A is affinely-independent and
   A2: B c=A;
  now let L be Linear_Combination of B such that
    A3: Sum L=0.V & sum L=0;
   L is Linear_Combination of A by A2,RLVECT_2:21;
   hence Carrier L={} by A1,A3,Th42;
  end;
  hence thesis by Th42;
 end;
