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;
reserve K for non void subset-closed SimplicialComplexStr;
reserve P for Function;

theorem Th52:
  dom P is with_non-empty_elements implies
    degree subdivision(P,KX) <= degree KX
  proof
  assume A1: dom P is with_non-empty_elements;
  set PP=subdivision(P,KX);
  per cases;
  suppose A2: KX is non finite-degree;
   KX is non void
   proof
    assume KX is void;
    then KX is empty-membered;
    hence thesis by A2;
   end;
   then degree KX=+infty by A2,Def12;
   hence thesis by XXREAL_0:3;
  end;
  suppose A3: KX is finite-degree;
   then reconsider d=degree KX as Integer;
   reconsider d1=d+1 as Nat by A3;
   for A be finite Subset of PP st A is simplex-like holds card A<=d+1
   proof
    let A be finite Subset of PP;
    assume A is simplex-like;
    then consider S be c=-linear finite simplex-like Subset-Family of KX such
that
     A4: A=P.:S by Def20;
    A5: A=P.:(S/\dom P) by A4,RELAT_1:112;
    per cases;
    suppose S/\dom P is empty;
     then 0<=d1 & A={} by A5;
     hence thesis;
    end;
    suppose A6: S/\dom P is non empty;
     reconsider uSP=union(S/\dom P) as Subset of KX;
     A7: S/\dom P c=S by XBOOLE_1:17;
     then union(S/\dom P) in S/\dom P by A6,Th9;
     then union(S/\dom P) in S by XBOOLE_0:def 4;
     then A8: uSP is simplex-like by TOPS_2:def 1;
     then A9: uSP in the topology of KX;
     the_family_of KX is finite-membered by A3,MATROID0:def 6;
     then reconsider uSP as finite Subset of KX by A9;
     KX is non void by A9,PENCIL_1:def 4;
     then card uSP<=d+1 by A8,Th25;
     then A10: Segm card uSP c= Segm d1 by NAT_1:39;
     card(S/\dom P)c=card union(S/\dom P) by A1,A7,Th10;
     then A11: card(S/\dom P)c=d1 by A10;
     card A c=card(S/\dom P) by A5,CARD_1:67;
     then Segm card A c= Segm d1 by A11;
     hence thesis by NAT_1:39;
    end;
   end;
   hence thesis by Th25;
  end;
 end;
