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 Th10:
  x in Int A implies ex L be Linear_Combination of A st L is convex & x = Sum L
  proof
    assume A1: x in Int A;
    then reconsider A1=A as non empty Subset of V;
    x in conv A by A1,Def1;
    then x in {Sum L where L is Convex_Combination of A1:L in ConvexComb(V)}
      by CONVEX3:5;
    then ex L be Convex_Combination of A1 st x=Sum L & L in ConvexComb(V);
    hence thesis;
  end;
