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;
reserve K for non void subset-closed SimplicialComplexStr;
reserve P for Function;

theorem Th58:
  subdivision(P,SX) is SubSimplicialComplex of subdivision(P,KX)
 proof
  set PS=subdivision(P,SX);
  set PK=subdivision(P,KX);
  A1: [#]SX c=[#]KX by Def13;
  A2: [#]PK=[#]KX by Def20;
  hence [#]PS c=[#]PK by A1,Def20;
  let x be object;
  assume x in the topology of PS;
  then reconsider A=x as Simplex of PS by PRE_TOPC:def 2;
  [#]PS=[#]SX by Def20;
  then reconsider B=A as Subset of PK by A1,A2,XBOOLE_1:1;
  consider SS be c=-linear finite simplex-like Subset-Family of SX such that
   A3: A=P.:SS by Def20;
  bool[#]SX c=bool[#]KX by A1,ZFMISC_1:67;
  then reconsider SK=SS as c=-linear finite Subset-Family of KX by XBOOLE_1:1;
  SK is simplex-like
  proof
   let C be Subset of KX;
   assume A4: C in SK;
   then reconsider c =C as Subset of SX;
   c is simplex-like by A4,TOPS_2:def 1;
   then A5: c in the topology of SX;
   the topology of SX c=the topology of KX by Def13;
   hence thesis by A5;
  end;
  then B is simplex-like by A3,Def20;
  hence thesis;
 end;
