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 Th60:
  for K1,K2 be SimplicialComplexStr of X st
      the TopStruct of K1 = the TopStruct of K2
    holds subdivision(P,K1) = subdivision(P,K2)
 proof
  let K1,K2 be SimplicialComplexStr of X such that
   A1: the TopStruct of K1=the TopStruct of K2;
  set P1=subdivision(P,K1),P2=subdivision(P,K2);
  A2: [#]K1=[#]P1 & [#]K2=[#]P2 by Def20;
  A3: the topology of P1 c=the topology of P2
  proof
   let x be object;
   assume A4: x in the topology of P1;
   then reconsider A=x as Subset of P1;
   reconsider A1=A as Subset of P2 by A1,A2;
   A is simplex-like by A4;
   then consider S be c=-linear finite simplex-like Subset-Family of K1 such
that
    A5: A=P.:S by Def20;
   reconsider S1=S as Subset-Family of K2 by A1;
   S c=the topology of K1 by Th14;
   then S1 is simplex-like by A1,Th14;
   then A1 is simplex-like by A5,Def20;
   hence thesis;
  end;
  the topology of P2 c=the topology of P1
  proof
   let x be object;
   assume A6: x in the topology of P2;
   then reconsider A=x as Subset of P2;
   reconsider A1=A as Subset of P1 by A1,A2;
   A is simplex-like by A6;
   then consider S be c=-linear finite simplex-like Subset-Family of K2 such
that
    A7: A=P.:S by Def20;
   reconsider S1=S as Subset-Family of K1 by A1;
   S c=the topology of K2 by Th14;
   then S1 is simplex-like by A1,Th14;
   then A1 is simplex-like by A7,Def20;
   hence thesis;
  end;
  hence thesis by A1,A2,A3,XBOOLE_0:def 10;
 end;
