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 Th7:
  |.Kr.| is empty iff Kr is empty-membered
 proof
  hereby assume A1: |.Kr.| is empty;
   assume Kr is with_non-empty_element;
   then the topology of Kr is with_non-empty_element;
   then consider x be non empty set such that
    A2: x in the topology of Kr;
   reconsider X=x as Subset of Kr by A2;
   (ex y being object st y in conv@X) & X is simplex-like by A2,XBOOLE_0:def 1
;
   hence contradiction by A1,Def3;
  end;
  assume A3: Kr is empty-membered;
  assume|.Kr.| is non empty;
  then consider x being object such that
   A4: x in |.Kr.|;
  consider A be Subset of Kr such that
   A5: A is simplex-like & x in conv@A by A4,Def3;
  A is non empty & A in the topology of Kr by A5;
  then the topology of Kr is with_non-empty_element;
  hence contradiction by A3;
 end;
