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
  for A be Subset of subdivision(P,KX) st
      dom P c=the topology of SX & A = [#]SX
    holds subdivision(P,SX)=subdivision(P,KX)|A
 proof
  set PK=subdivision(P,KX);
  reconsider PS=subdivision(P,SX) as strict SubSimplicialComplex of PK by Th58;
  let A be Subset of subdivision(P,KX) such that
   A1: dom P c=the topology of SX and
   A2: A=[#]SX;
  now let a be Subset of PS;
   assume a in the topology of PK;
   then reconsider b=a as Simplex of PK by PRE_TOPC:def 2;
   consider SS be c=-linear finite simplex-like Subset-Family of KX such that
    A3: b=P.:SS by Def20;
   SS/\dom P c=dom P by XBOOLE_1:17;
   then A4: SS/\dom P c=the topology of SX by A1;
   SS/\dom P c=SS by XBOOLE_1:17;
   then reconsider Sd=SS/\dom P as c=-linear finite Subset-Family of SX by A4,
XBOOLE_1:1;
   A5: Sd is simplex-like
   proof
    let B be Subset of SX;
    assume B in Sd;
    then B in dom P by XBOOLE_0:def 4;
    hence thesis by A1;
   end;
   P.:SS=P.:Sd by RELAT_1:112;
   hence a is simplex-like by A3,A5,Def20;
  end;
  then A6: PS is maximal;
  [#]PS=[#]SX by Def20;
  hence thesis by A2,A6,Def16;
 end;
