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
  for S,A be Subset of RLS st A c= conv S holds conv S = conv(S\/A)
  proof
    let S,A be Subset of RLS such that
    A1: A c=conv S;
    thus conv S c=conv(S\/A) by Th3,XBOOLE_1:7;
    S c=conv S by Lm1;
    then S\/A c=conv S by A1,XBOOLE_1:8;
    hence thesis by CONVEX1:30;
  end;
