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;

theorem Th22:
  degree K = -1 iff K is empty-membered
 proof
  per cases;
  suppose K is void;
   hence thesis by Def12;
  end;
  suppose A1: K is non void;
   hereby assume A2: degree K=-1;
    then A3: K is finite-degree by A1,Def12;
    assume K is with_non-empty_element;
    then the topology of K is with_non-empty_element;
    then consider S be non empty set such that
     A4: S in the topology of K;
    reconsider S as Subset of K by A4;
    A5: S is simplex-like by A4;
    then reconsider S as finite Subset of K by A1,A3;
    card S<=-1+1 by A1,A2,A3,A5,Def12;
    then card S=0;
    hence contradiction;
   end;
   assume A6: K is empty-membered;
   then consider S be Subset of K such that
    A7: S is simplex-like and
    A8: card S=degree K+1 by A1,Def12;
   A9: S in the topology of K by A7;
   assume degree K<>-1;
   then card S<>-1+1 by A8,XXREAL_3:11;
   then A10: S is non empty;
   the topology of K is empty-membered by A6;
   hence thesis by A9,A10;
  end;
 end;
