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
  subdivision(n,P,SX) is SubSimplicialComplex of subdivision(n,P,KX)
 proof
  defpred P[Nat] means
   subdivision($1,P,SX) is SubSimplicialComplex of subdivision($1,P,KX);
  A1: for n st P[n] holds P[n+1]
  proof
   let n;
   assume P[n];
   then reconsider Pn=subdivision(n,P,SX) as
     SubSimplicialComplex of subdivision(n,P,KX);
   A2: n in NAT by ORDINAL1:def 12;
   then A3: subdivision(n+1,P,SX)=subdivision(1,P,Pn) by Th63
    .=subdivision(P,Pn) by Th62;
   subdivision(n+1,P,KX)=subdivision(1,P,subdivision(n,P,KX)) by A2,Th63
    .=subdivision(P,subdivision(n,P,KX)) by Th62;
   hence thesis by A3,Th58;
  end;
  subdivision(0,P,SX)=SX by Th61;
  then A4: P[0] by Th61;
  for n holds P[n] from NAT_1:sch 2(A4,A1);
  hence thesis;
 end;
