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 Th3:
  for A,B be Subset of RLS st A c= B holds conv A c= conv B
  proof
    let A,B be Subset of RLS such that
    A1: A c=B;
    A2: Convex-Family B c=Convex-Family A
    proof
      let x be object;
      assume A3: x in Convex-Family B;
      then reconsider X=x as Subset of RLS;
      B c=X by A3,CONVEX1:def 4;
      then A4: A c=X by A1;
      X is convex by A3,CONVEX1:def 4;
      hence thesis by A4,CONVEX1:def 4;
    end;
    [#]RLS is convex;
    then [#]RLS in Convex-Family B by CONVEX1:def 4;
    then A5: meet(Convex-Family A)c=meet(Convex-Family B) by A2,SETFAM_1:6;
    let y be object;
    assume y in conv A;
    hence thesis by A5;
  end;
