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;

theorem
  for K be subset-closed SimplicialComplexStr of V holds
    x in |.K.| iff ex A be Subset of K st A is simplex-like & x in Int @A
 proof
  let K be subset-closed SimplicialComplexStr of V;
  hereby assume x in |.K.|;
   then consider A be Subset of K such that
    A1: A is simplex-like and
    A2: x in conv@A by Def3;
   conv@A=union{Int B where B is Subset of V:B c=@A} by RLAFFIN2:8;
   then consider IB be set such that
    A3: x in IB and
    A4: IB in {Int B where B is Subset of V:B c=@A} by A2,TARSKI:def 4;
   consider B be Subset of V such that
    A5: IB=Int B and
    A6: B c=@A by A4;
   reconsider B1=B as Subset of K by A6,XBOOLE_1:1;
   take B1;
   A in the topology of K by A1;
   then K is non void by PENCIL_1:def 4;
   hence B1 is simplex-like & x in Int@B1 by A1,A3,A5,A6,MATROID0:1;
  end;
  given A be Subset of K such that
   A7: A is simplex-like and
   A8: x in Int@A;
  x in conv@A by A8,RLAFFIN2:def 1;
  hence thesis by A7,Def3;
 end;
