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
  dom P/\the topology of KX c= Y implies
    subdivision(P|Y,KX) = subdivision(P,KX)
 proof
  set PX=subdivision(P|Y,KX);
  set PP=subdivision(P,KX);
  A1: (P|Y)|dom(P|Y)=P|(dom(P|Y)) by RELAT_1:58,74;
  A2: [#]PX=[#]KX & [#]PP=[#]KX by Def20;
  assume A3: dom P/\the topology of KX c=Y;
  A4: the topology of PP c=the topology of PX
  proof
   let x be object;
   assume x in the topology of PP;
   then reconsider A=x as Simplex of PP by PRE_TOPC:def 2;
   reconsider B=A as Subset of PX by A2;
   consider S be c=-linear finite simplex-like Subset-Family of KX such that
    A5: A=P.:S by Def20;
   A6: S/\dom P c=Y
   proof
    let x be object;
    assume A7: x in S/\dom P;
    then reconsider A=x as Subset of KX;
    x in S by A7,XBOOLE_0:def 4;
    then A is simplex-like by TOPS_2:def 1;
    then A8: A in the topology of KX;
    x in dom P by A7,XBOOLE_0:def 4;
    then A in (the topology of KX)/\dom P by A8,XBOOLE_0:def 4;
    hence thesis by A3;
   end;
   then A9: S/\dom P/\Y=S/\dom P by XBOOLE_1:28;
   B=P.:(S/\dom P) by A5,RELAT_1:112
    .=(P|Y).:(S/\dom P/\Y) by A6,A9,RELAT_1:129
    .=(P|Y).:(S/\(dom P/\Y)) by XBOOLE_1:16
    .=(P|Y).:(S/\dom(P|Y)) by RELAT_1:61
    .=(P|Y).:S by RELAT_1:112;
   then B is simplex-like by Def20;
   hence thesis;
  end;
  P|dom P=P & P|Y=(P|Y)|dom(P|Y);
  then PX is SubSimplicialComplex of PP by A1,Th54,RELAT_1:60;
  then the topology of PX c=the topology of PP by Def13;
  hence thesis by A2,A4,XBOOLE_0:def 10;
 end;
