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 Th51:
  degree subdivision(P,KX) <= degree KX + 1
  proof
  set PP=subdivision(P,KX);
  per cases;
  suppose KX is void;
   then KX is empty-membered & degree PP=-1 by Th22;
   hence thesis;
  end;
  suppose A1: KX is non void non finite-degree;
   A2: degree PP<=+infty & degree KX+0<=degree KX+1 by XXREAL_0:3,XXREAL_3:36;
   degree KX+0=degree KX & degree KX=+infty by A1,Def12,XXREAL_3:4;
   hence thesis by A2,XXREAL_0:2;
  end;
  suppose A3: KX is non void 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)+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;
    set Sd=S/\dom P;
    A=P.:(S/\dom P) by A4,RELAT_1:112;
    then Segm card A c= Segm card Sd by CARD_1:67;
    then A5: card A<=card Sd by NAT_1:39;
    Sd\{{}}c=S by XBOOLE_1:18,36;
    then reconsider SP=Sd\{{}} as c=-linear finite simplex-like Subset-Family
of KX by TOPS_2:11;
    SP\/{{}}=Sd\/{{}} by XBOOLE_1:39;
    then A6: card{{}}=1 & card Sd<=card(SP\/{{}}) by CARD_1:30,NAT_1:43
,XBOOLE_1:7;
    card(SP\/{{}})<=card SP+card{{}} by CARD_2:43;
    then A7: card Sd<=card SP+1 by A6,XXREAL_0:2;
    per cases;
    suppose A8: SP is empty;
     0+1<=d1+1 by XREAL_1:6;
     then card Sd<=d1+1 by A6,A8,XXREAL_0:2;
     hence thesis by A5,XXREAL_0:2;
    end;
    suppose A9: SP is non empty;
     reconsider uSP=union SP as Subset of KX;
     union SP in SP by A9,Th9;
     then A10: uSP is simplex-like by TOPS_2:def 1;
     not{} in SP by ZFMISC_1:56;
     then SP is with_non-empty_elements;
     then A11: card SP c=card union SP by Th10;
     reconsider uSP as finite Subset of KX by A3;
     card uSP<=d1 by A3,A10,Th25;
     then Segm card uSP c= Segm d1 by NAT_1:39;
     then Segm card SP c= Segm d1 by A11;
     then card SP<=d1 by NAT_1:39;
     then card SP+1<=d1+1 by XREAL_1:6;
     then card Sd<=d1+1 by A7,XXREAL_0:2;
     hence thesis by A5,XXREAL_0:2;
    end;
   end;
   hence thesis by Th25;
  end;
 end;
