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;

theorem Th32:
  degree SX <= degree KX
 proof
  per cases;
  suppose A1: SX is non finite-degree;
   SX is non void
   proof
    assume SX is void;
    then SX is empty-membered;
    hence contradiction by A1;
   end;
   then KX is non finite-degree non void & degree SX=+infty by A1,Def12;
   hence thesis by Def12;
  end;
  suppose SX is void;
   then degree SX=-1 by Def12;
   hence thesis by Th23;
  end;
  suppose A2: SX is non void finite-degree;
   then A3: KX is non void;
   per cases;
   suppose KX is non finite-degree;
    then degree KX=+infty by A3,Def12;
    hence thesis by XXREAL_0:3;
   end;
   suppose A4: KX is finite-degree;
    A5: the topology of SX c=the topology of KX by Def13;
    consider S be Subset of SX such that
     A6: S is simplex-like and
     A7: card S=degree SX+1 by A2,Def12;
    A8: S in the topology of SX by A6;
    then S in the topology of KX by A5;
    then reconsider S1=S as finite Subset of KX by A4,A7;
    S1 is simplex-like by A5,A8;
    then degree SX+1<=degree KX+1 by A3,A4,A7,Def12;
    then A9: degree SX+1-1<=degree KX+1-1 by XXREAL_3:37;
    degree SX+1-1=degree SX by XXREAL_3:24;
    hence thesis by A9,XXREAL_3:24;
   end;
  end;
 end;
