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
  holds degree Skeleton_of(KX,i) <= degree KX
 proof let i be dim-like number;
  set S=Skeleton_of(KX,i);
  per cases;
  suppose KX is void or S is void;
   then KX is empty-membered or S is empty-membered;
   then degree Skeleton_of(KX,i)=-1 by Th22;
   hence thesis by Th23;
  end;
  suppose A1: KX is non void finite-degree & S is non void;
   then reconsider d=degree KX as Integer;
   now let s be finite Subset of S;
    assume s is simplex-like;
    then s in subset-closed_closure_of the_subsets_with_limited_card
    (Segm(i+1),the topology of KX);
    then consider a be set such that
     A2: s c=a and
     A3: a in the_subsets_with_limited_card(Segm(i+1),the topology of KX)
     by Th2;
    A4: a in the topology of KX by A3,Def2;
    reconsider a as finite Subset of KX by A3;
    Segm card s c= Segm card a by A2,CARD_1:11;
    then A5: card s<=card a by NAT_1:39;
    a is simplex-like & KX is non void by A4,PENCIL_1:def 4;
    then card a<=d+1 by Th25;
    hence card s<=d+1 by A5,XXREAL_0:2;
   end;
   hence thesis by A1,Th25;
  end;
  suppose A6: KX is non finite-degree;
   KX is non void
   proof
    assume KX is void;
    then KX is empty-membered;
    hence thesis by A6;
   end;
   then degree KX=+infty by A6,Def12;
   hence thesis by XXREAL_0:3;
  end;
 end;
