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
 P.:(the topology of KX) c= Y implies subdivision(Y|`P,KX) = subdivision(P,KX)
 proof
  set PK=subdivision(P,KX);
  P=rng P|`P & Y|`(rng P|`P)=(Y/\rng P)|`P by RELAT_1:96;
  then reconsider PY=subdivision(Y|`P,KX) as SubSimplicialComplex of PK
    by Th56,XBOOLE_1:17;
  A1: [#]PY=[#]KX & [#]PK=[#]KX by Def20;
  assume A2: P.:(the topology of KX)c=Y;
  A3: the topology of PK c=the topology of PY
  proof
   let x be object;
   assume x in the topology of PK;
   then reconsider A=x as Simplex of PK by PRE_TOPC:def 2;
   consider S be c=-linear finite simplex-like Subset-Family of KX such that
    A4: A=P.:S by Def20;
   reconsider A as Subset of PY by A1;
   S c=the topology of KX
   proof
    let y be object such that
     A5: y in S;
    reconsider y as Subset of KX by A5;
    y is simplex-like by A5,TOPS_2:def 1;
    hence thesis;
   end;
   then A c=P.:(the topology of KX) by A4,RELAT_1:123;
   then A/\Y=A by A2,XBOOLE_1:1,28;
   then A=Y|`P.:S by A4,FUNCT_1:67;
   then A is simplex-like by Def20;
   hence thesis;
  end;
  the topology of PY c=the topology of PK by Def13;
  hence thesis by A1,A3,XBOOLE_0:def 10;
 end;
