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 Th62:
  subdivision(1,P,KX) = subdivision(P,KX)
 proof
  consider F be Function such that
   A1: F.0=KX and
   A2: F.1=subdivision(1,P,KX) and
   dom F=NAT and
   A3: for k for KX1 be SimplicialComplexStr of X st KX1=F.k holds F.(k+1)=
subdivision(P,KX1) by Def21;
  F.(0+1)=subdivision(P,KX) by A1,A3;
  hence thesis by A2;
 end;
