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 Th19:
  (center_of_mass V).If in If iff card If=1
  proof
    set B=center_of_mass V;
    hereby assume A1: B.If in If;
      then reconsider BA=B.If as Element of V;
      B.If in conv If by A1,Th16;
      then 1=(BA|--If).BA by A1,RLAFFIN1:72
      .=1/card If by A1,Th18;
      hence 1=card If by XCMPLX_1:58;
    end;
    assume A2: card If=1;
    then consider x being object such that
    A3: {x}=If by CARD_2:42;
    x in If by A3,TARSKI:def 1;
    then reconsider x as Element of V;
    (center_of_mass V).If=1/card If*Sum If by A3,Def2
    .=1/1*x by A2,A3,RLVECT_2:9
    .=x by RLVECT_1:def 8;
    hence thesis by A3,TARSKI:def 1;
  end;
