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 Th26:
  As is simplex-like & Bs is simplex-like & Int@As meets conv@Bs
    implies As c= Bs
 proof
  assume that
   A1: As is simplex-like and
   A2: Bs is simplex-like and
   A3: Int@As meets conv@Bs;
  consider x being object such that
   A4: x in Int@As and
   A5: x in conv@Bs by A3,XBOOLE_0:3;
  x in union{Int b where b is Subset of V:b c=@Bs} by A5,RLAFFIN2:8;
  then consider Ib be set such that
   A6: x in Ib and
   A7: 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
   A8: Ib=Int b and
   A9: b c=@Bs by A7;
  reconsider b1=b as Subset of Ks by A9,XBOOLE_1:1;
  As in the topology of Ks by A1;
  then Ks is non void by PENCIL_1:def 4;
  then A10: b1 is simplex-like by A2,A9,MATROID0:1;
  Int@As meets Int@b1 by A4,A6,A8,XBOOLE_0:3;
  hence thesis by A1,A9,A10,Th25;
 end;
