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 -1 <= i & Skeleton_of(KX,i) = the TopStruct of KX
  holds degree KX <= i
 proof let i be dim-like number;
  assume A1: -1<=i;
  per cases;
  suppose KX is empty-membered;
   hence thesis by A1,Th22;
  end;
  suppose A2: KX is with_non-empty_element;
   reconsider i1=i+1 as Element of NAT by INT_1:3;
   assume A3: Skeleton_of(KX,i)=the TopStruct of KX;
   A4: now let S be finite Subset of KX;
    assume S is simplex-like;
    then S in subset-closed_closure_of the_subsets_with_limited_card(i1,the
topology of KX) by A3;
    then consider y such that
     A5: S c=y & y in the_subsets_with_limited_card(i1,the topology of KX)
     by Th2;
    card S c=card y & card y c=card i1 by A5,Def2,CARD_1:11;
    then A6: card S c=card Segm i1;
    card S=card Segm card S;
    hence card S<=i1 by A6,NAT_1:40;
   end;
   for x st x in the topology of KX holds x is finite by A3;
   then the_family_of KX is finite-membered;
   then KX is finite-membered;
   hence thesis by A2,A4,Th25;
  end;
 end;
