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
  A c= If & (center_of_mass V).If in Affin A implies If = A
  proof
    set B=center_of_mass V;
    assume that
    A1: A c=If and
    A2: B.If in Affin A;
    A3: B.If|--If=B.If|--A by A1,A2,RLAFFIN1:77;
    reconsider i=If as finite set;
    assume If<>A;
    then not If c=A by A1;
    then consider x being object such that
    A4: x in If and
    A5: not x in A;
    reconsider x as Element of V by A4;
    A6: (B.If|--If).x=1/(card i) by A4,Th18;
    Carrier(B.If|--A)c=A by RLVECT_2:def 6;
    then not x in Carrier(B.If|--A) by A5;
    hence contradiction by A3,A4,A6;
  end;
