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
  for A,B be finite Subset of V st
      A is affinely-independent & Affin A c= Affin B & card A = card B
    holds B is affinely-independent
 proof
  let A,B be finite Subset of V such that
   A1: A is affinely-independent & Affin A c=Affin B & card A=card B;
  {}V c=B;
  then consider Ib be affinely-independent Subset of V such that
   {}V c=Ib and
   A2: Ib c=B and
   A3: Affin Ib=Affin B by Th60;
  reconsider IB=Ib as finite Subset of V by A2;
  A4: card IB<=card B by A3,Th79;
  card B<=card IB by A1,A3,Th79;
  hence thesis by A2,A4,CARD_2:102,XXREAL_0:1;
 end;
