reserve x,y for set,
        r,s for Real,
        n for Nat,
        V for RealLinearSpace,
        v,u,w,p for VECTOR of V,
        A,B for Subset of V,
        Af for finite Subset of V,
        I for affinely-independent Subset of V,
        If for finite affinely-independent Subset of V,
        F for Subset-Family of V,
        L1,L2 for Linear_Combination of V;

theorem Th20:
  If is non empty implies (center_of_mass V).If in Int If
  proof
    set BA=(center_of_mass V).If|--If;
    A1: If c=Carrier BA
    proof
      let x be object;
      assume A2: x in If;
      then BA.x=1/card If by Th18;
      hence thesis by A2;
    end;
    assume If is non empty;
    then A3: (center_of_mass V).If in conv If by Th16;
    Carrier BA c=If by RLVECT_2:def 6;
    then Carrier BA=If & BA is convex by A1,A3,RLAFFIN1:71;
    then conv If c=Affin If & Sum BA in Int If by Th12,RLAFFIN1:65;
    hence thesis by A3,RLAFFIN1:def 7;
  end;
