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 Th67:
  Affin conv A = Affin A
 proof
  thus Affin conv A c=Affin A by Th51,Th65;
  let x be object;
  assume x in Affin A;
  then x in {Sum L where L is Linear_Combination of A:sum L=1} by Th59;
  then consider L be Linear_Combination of A such that
   A1: x=Sum L and
   A2: sum L=1;
  reconsider K=L as Linear_Combination of conv A by Lm1,RLVECT_2:21;
  Sum K in {Sum M where M is Linear_Combination of conv A:sum M=1} by A2;
  hence thesis by A1,Th59;
 end;
