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 Th17:
  union F is finite implies (center_of_mass V).:F c= conv union F
  proof
    set B=center_of_mass V;
    assume A1: union F is finite;
    let y be object;
    assume y in B.:F;
    then consider x being object such that
    A2: x in dom B and
    A3: x in F and
    A4: B.x=y by FUNCT_1:def 6;
    reconsider x as non empty Subset of V by A2,ZFMISC_1:56;
    x c=union F by A3,ZFMISC_1:74;
    then A5: y in conv x by A1,A4,Th16;
    conv x c=conv union F by A3,RLTOPSP1:20,ZFMISC_1:74;
    hence thesis by A5;
  end;
