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 Th63:
  for n,k be Element of NAT holds
    subdivision(n+k,P,KX) = subdivision(n,P,subdivision(k,P,KX))
 proof
  let n,k be Element of NAT;
  consider Fn be Function such that
   A1: Fn.0=subdivision(k,P,KX) and
   A2: Fn.n=subdivision(n,P,subdivision(k,P,KX)) and
   dom Fn=NAT and
   A3: for m be Nat for KX1 be SimplicialComplexStr of X st KX1=Fn.m holds Fn.(
m+1)=subdivision(P,KX1) by Def21;
  consider Fnm be Function such that
   A4: Fnm.0=KX and
   A5: Fnm.(n+k)=subdivision(n+k,P,KX) and
   dom Fnm=NAT and
   A6: for m be Nat for KX1 be SimplicialComplexStr of X st KX1=Fnm.m holds Fnm
.(m+1)=subdivision(P,KX1) by Def21;
  defpred R[Nat] means
   $1<=n implies Fn.$1=Fnm.($1+k) & ex SX be SimplicialComplexStr of X st Fn.$1
=SX;
  A7: for m be Nat st R[m] holds R[m+1]
  proof
   let m be Nat such that
    A8: R[m];
   assume A9: m+1<=n;
   then consider SX be SimplicialComplexStr of X such that
    A10: Fn.m=SX by A8,NAT_1:13;
   subdivision(P,SX)=Fnm.(m+k+1) by A6,A8,A9,A10,NAT_1:13;
   hence thesis by A3,A10;
  end;
  consider Fm be Function such that
   A11: Fm.0=KX and
   A12: Fm.k=subdivision(k,P,KX) and
   dom Fm=NAT and
   A13: for m be Nat for KX1 be SimplicialComplexStr of X st KX1=Fm.m holds Fm.
(m+1)=subdivision(P,KX1) by Def21;
  defpred P[Nat] means
   $1<=k implies Fm.$1=Fnm.$1 & ex SX be SimplicialComplexStr of X st Fm.$1=SX;
  A14: for m be Nat st P[m] holds P[m+1]
  proof
   let m be Nat such that
    A15: P[m];
   assume A16: m+1<=k;
   then consider SX be SimplicialComplexStr of X such that
    A17: Fm.m=SX by A15,NAT_1:13;
   subdivision(P,SX)=Fnm.(m+1) by A6,A15,A16,A17,NAT_1:13;
   hence thesis by A13,A17;
  end;
  A18: P[0 qua Nat] by A4,A11;
  for k holds P[k] from NAT_1:sch 2(A18,A14);
  then A19: R[0 qua Nat] by A1,A12;
  for k holds R[k] from NAT_1:sch 2(A19,A7);
  hence thesis by A2,A5;
 end;
