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 Th54:
  Y c=Z implies
    subdivision(P|Y,KX) is SubSimplicialComplex of subdivision(P|Z,KX)
 proof
  set PY=subdivision(P|Y,KX);
  set PZ=subdivision(P|Z,KX);
  A1: dom(P|Z)=Z/\dom P by RELAT_1:61;
  assume A2: Y c=Z;
  then Y/\Z=Y by XBOOLE_1:28;
  then A3: dom(P|Y)=(Z/\Y)/\dom P by RELAT_1:61
   .=Y/\dom(P|Z) by A1,XBOOLE_1:16;
  A4: [#]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;
  [#]PZ=[#]KX by Def20;
  then reconsider B=A as Subset of PZ by A4;
  consider S be c=-linear finite simplex-like Subset-Family of KX such that
   A5: A=(P|Y).:S by Def20;
  S/\Y c=S by XBOOLE_1:17;
  then reconsider SY=S/\Y as c=-linear finite simplex-like Subset-Family of KX
by TOPS_2:11;
  A6: dom(P|Y)c=Y & dom(P|Y)/\S c=dom(P|Y) by RELAT_1:58,XBOOLE_1:17;
  then A7: dom(P|Y)/\S c=Y;
  B=(P|Y).:(dom(P|Y)/\S) by A5,RELAT_1:112
   .=P.:(dom(P|Y)/\S) by A6,RELAT_1:129,XBOOLE_1:1
   .=(P|Z).:(dom(P|Y)/\S) by A2,A7,RELAT_1:129,XBOOLE_1:1
   .=(P|Z).:(dom(P|Z)/\SY) by A3,XBOOLE_1:16
   .=(P|Z).:SY by RELAT_1:112;
  then B is simplex-like by Def20;
  hence thesis;
 end;
