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
 holds degree Skeleton_of(KX,i) <= i
 proof let i be dim-like number;
  set swlc=the_subsets_with_limited_card(Segm(i+1),the topology of KX);
  set S=Skeleton_of(KX,i);
  assume A1: -1<=i;
  reconsider i1=i+1 as Element of NAT by INT_1:3;
  now let A be finite Subset of S;
   assume A is simplex-like;
   then A in the topology of S;
   then consider x such that
    A2: A c=x & x in swlc by Th2;
   card x c=card Segm (i+1) & card A c=card x by A2,Def2,CARD_1:11;
   then A3: card A c=card Segm i1;
   card Segm card A=card A;
   hence card A<=i+1 by A3,NAT_1:40;
  end;
  hence thesis by A1,Th25;
 end;
