reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;
reserve K for SimplicialComplexStr;
reserve KX for SimplicialComplexStr of X,
        SX for SubSimplicialComplex of KX;
reserve SC for SimplicialComplex of X;

theorem
  SC|[#]SC = the TopStruct of SC
 proof
  set T=the TopStruct of SC;
  A1: [#]T c=[#]SC & bool[#]T/\the topology of SC=the topology of SC by
XBOOLE_1:28;
  the_family_of SC=the_family_of T & the_family_of SC is finite-membered
subset-closed by MATROID0:def 6;
  then A2: T is finite-membered subset-closed;
  [#]SC c=X by Def9;
  then [#]T c=X;
  then T is SimplicialComplexStr of X by Def9;
  then reconsider T as maximal SubSimplicialComplex of SC by A1,A2,Def13,Th33;
  [#]T=[#]SC;
  hence thesis by Def16;
 end;
