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 Th56:
  Y c= Z implies
    subdivision(Y|`P,KX) is SubSimplicialComplex of subdivision(Z|`P,KX)
 proof
  assume A1: Y c=Z;
  set PZ=subdivision(Z|`P,KX);
  set PY=subdivision(Y|`P,KX);
  A2: [#]PY=[#]KX by Def20;
  hence [#]PY c=[#]PZ by Def20;
  let x be object;
  assume x in the topology of PY;
  then reconsider A=x as Simplex of PY by PRE_TOPC:def 2;
  consider S be c=-linear finite simplex-like Subset-Family of KX such that
   A3: A=Y|`P.:S by Def20;
  S/\dom(Y|`P)c=S by XBOOLE_1:17;
  then reconsider Sd=S/\dom(Y|`P)
    as c=-linear finite simplex-like Subset-Family of KX by TOPS_2:11;
  Y|`(Z|`P)=Y|`P by A1,RELAT_1:99;
  then A4: (Y|`P).:Sd c=(Z|`P).:Sd by RELAT_1:86,124;
  [#]PZ=[#]KX by Def20;
  then reconsider A as Subset of PZ by A2;
  A5: Z|`P.:Sd c=Y|`P.:Sd
  proof
   let y be object;
   assume y in (Z|`P).:Sd;
   then consider x being object such that
    A6: x in dom(Z|`P) and
    A7: x in Sd and
    A8: (Z|`P).x=y by FUNCT_1:def 6;
   A9: x in dom(Y|`P) by A7,XBOOLE_0:def 4;
   P.x=y by A6,A8,FUNCT_1:53;
   then (Y|`P).x=y by A9,FUNCT_1:53;
   hence thesis by A7,A9,FUNCT_1:def 6;
  end;
  A=Y|`P.:Sd by A3,RELAT_1:112;
  then A=Z|`P.:Sd by A4,A5;
  then A is simplex-like by Def20;
  hence thesis;
 end;
