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
  Int A is non empty implies A is finite
  proof
    assume Int A is non empty;
    then consider x being object such that
    A1: x in Int A;
    consider L be Linear_Combination of A such that
   A2: L is convex & x=Sum L by A1,Th10;
   Carrier L=A by A1,A2,Th11;
   hence A is finite;
 end;
