reserve x,y,X for set,
        r for Real,
        n,k for Nat;
reserve RLS for non empty RLSStruct,
        Kr,K1r,K2r for SimplicialComplexStr of RLS,
        V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve Ks for simplex-join-closed SimplicialComplex of V,
        As,Bs for Subset of Ks,
        Ka for non void affinely-independent SimplicialComplex of V,
        Kas for non void affinely-independent simplex-join-closed
                 SimplicialComplex of V,
        K for non void affinely-independent simplex-join-closed total
                 SimplicialComplex of V;

theorem
  As is simplex-like & @As is affinely-independent & Bs is simplex-like
    implies (Int@As c= conv @Bs iff As c= Bs)
 proof
  assume that
   A1: As is simplex-like and
   A2: @As is affinely-independent and
   A3: Bs is simplex-like;
  As in the topology of Ks by A1;
  then A4: Ks is non void by PENCIL_1:def 4;
  per cases;
  suppose A5: As is empty;
   thus thesis by A5;
  end;
  suppose As is non empty;
   then Int@As is non empty by A1,A2,A4,RLAFFIN2:20;
   then consider x being object such that
    A6: x in Int@As;
   hereby assume Int@As c=conv@Bs;
    then x in conv@Bs by A6;
    then x in union{Int b where b is Subset of V:b c=@Bs} by RLAFFIN2:8;
    then consider Ib be set such that
     A7: x in Ib and
     A8: Ib in {Int b where b is Subset of V:b c=@Bs} by TARSKI:def 4;
    consider b be Subset of V such that
     A9: Ib=Int b and
     A10: b c=@Bs by A8;
    reconsider b1=b as Subset of Ks by A10,XBOOLE_1:1;
    A11: b1 is simplex-like by A3,A4,A10,MATROID0:1;
    Int@As meets Int@b1 by A6,A7,A9,XBOOLE_0:3;
    hence As c=Bs by A1,A10,A11,Th25;
   end;
   assume As c=Bs;
   then Int@As c=conv@As & conv@As c=conv@Bs by RLAFFIN1:3,RLAFFIN2:5;
   hence thesis;
  end;
 end;
