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;

theorem
 for i being dim-like number
  st KX is subset-closed & degree KX <= i
  holds Skeleton_of(KX,i) = the TopStruct of KX
 proof let i be dim-like number;
  assume that
   A1: KX is subset-closed and
   A2: degree KX<=i;
  set S=Skeleton_of(KX,i);
  i in REAL by XREAL_0:def 1;
  then degree KX<+infty by A2,XXREAL_0:2,9;
  then A3: KX is finite-degree or KX is empty-membered by Def12;
  then A4: the_family_of KX is finite-membered by MATROID0:def 6;
  A5: the topology of KX c=the topology of S
  proof
   let x be object;
   A6: degree KX+1<=i+1 by A2,XXREAL_3:36;
   assume A7: x in the topology of KX;
   then reconsider A=x as finite Subset of KX by A4;
   A is simplex-like & KX is non void by A7,PENCIL_1:def 4;
   then card A<=degree KX+1 by A3,Def12;
   then card A<=i+1 & i+1 in NAT by A6,INT_1:3,XXREAL_0:2;
   then card Segm card A c=card Segm(i+1) by NAT_1:40;
   then A in the_subsets_with_limited_card(Segm(i+1),the topology of KX)
   by A7,Def2;
   hence thesis by Th2;
  end;
  A8: the_subsets_with_limited_card(Segm(i+1),the topology of KX)c=
  the topology of KX by Def2;
  the_family_of KX is subset-closed by A1;
  then the topology of S c=the topology of KX by A8,Th3;
  hence thesis by A5,XBOOLE_0:def 10;
 end;
