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 Th25:
 K is non void or i >= -1 implies (degree K <= i iff K is finite-membered &
   for S be finite Subset of K st S is simplex-like holds card S <= i+1)
 proof
  assume A1: K is non void or i>=-1;
  per cases;
  suppose A2: K is void;
    then
   A3: for S be finite Subset of K st S is simplex-like holds
    card S<=i+1 by PENCIL_1:def 4;
   K is empty-membered by A2;
   hence thesis by A1,A2,A3,Th22;
  end;
  suppose A4: K is non void;
   hereby assume A5: degree K<=i;
    then A6: degree K+1<=i+1 by XXREAL_3:35;
    i in REAL by XREAL_0:def 1;
    then A7: degree K<>+infty by A5,XXREAL_0:9;
    then K is finite-degree or K is empty-membered by Def12;
    hence K is finite-membered;
    let S be finite Subset of K;
    assume A8: S is simplex-like;
    K is non void finite-degree or K is void by A7,Def12;
    then card S<=degree K+1 by A4,A8,Def12;
    hence card S<=i+1 by A6,XXREAL_0:2;
   end;
   assume that
    A9: K is finite-membered and
    A10: for S be finite Subset of K st S is simplex-like holds card S<=i+1;
   consider S be object such that
    A11: S in the topology of K by A4,XBOOLE_0:def 1;
   reconsider S as Subset of K by A11;
   A12: S is simplex-like by A11;
   then reconsider S as finite Subset of K by A9;
   card S<=i+1 by A10,A12;
   then reconsider i1=i+1 as Element of NAT by INT_1:3;
   for A be finite Subset of K st A is open holds card A<=i1 by A10;
   then A13: K is finite-degree by A9;
   then reconsider d=degree K as Integer;
   ex S1 be Subset of K st S1 is simplex-like & card S1=degree K+1 by A4,A13
,Def12;
   then d+1<=i+1 by A9,A10;
   hence thesis by XREAL_1:6;
  end;
 end;
