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 Th65:
  conv A c= Affin A
 proof
  let x be object;
  assume A1: x in conv A;
  then reconsider A1=A as non empty Subset of V;
  conv(A1)={Sum(L) where L is Convex_Combination of A1:L in ConvexComb(V)}
by CONVEX3:5;
  then consider L be Convex_Combination of A1 such that
   A2: Sum L=x and
   L in ConvexComb(V) by A1;
  sum L=1 by Th62;
  then x in {Sum K where K is Linear_Combination of A:sum K=1} by A2;
  hence thesis by Th59;
 end;
