<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>;<br/><b>let </b><font color="Maroon" title="c2">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">k</font> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green" title="E79">A2</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>and </b><br/><a NAME="E3:97"/><i><font color="Green" title="E80">A3</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>set </b><font color="Maroon" title="c6">iseq</font> =  <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font>;<br/>
<a NAME="E4:97"/>
(  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> &amp; <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K7" title="POLYFORM:func.7">dim</a> <font color="Maroon" title="c1">p</font> )
 
<b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:97"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="polyform.html#K15" title="POLYFORM:func.15">-th-polytope</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font>
 <b>and </b><br/><a NAME="E7:97"/><i><font color="Green" title="E82">A5</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font>
 <b>and </b><br/><a NAME="E8:97"/><i><font color="Green" title="E83">A6</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font>
 <b>by </b><i><a class="ref" href="polyform.html#T33" target="_self" title="POLYFORM:th.33">Th33</a></i>;<br/>
<a NAME="E9:97"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="ref" href="polyform.html#D5" target="_self" title="POLYFORM:def.5">Def5</a></i>;<br/>
<a NAME="E10:97"/><b>then </b><i><font color="Green" title="E84">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyform.html#K11" title="POLYFORM:func.11">num-polytopes</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">k</font>
 
<b>by </b><i><a class="ref" href="polyform.html#D16" target="_self" title="POLYFORM:def.16">Def16</a></i>;<br/>
<a NAME="E11:97"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:97"/><b>then </b><i><font color="Green" title="E85">A8</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E7:97"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="polyform.html#E8:97"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="polyform.html#E10:97"><i><font color="Green" title="E84">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:97"/><i><font color="Green" title="E86">A9</font></i>: 
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="polyform.html#E2:97"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="polyform.html#E3:97"><i><font color="Green" title="E80">A3</font></i></a>, <a class="txt" href="polyform.html#E6:97"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="polyform.html#E7:97"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="polyform.html#E8:97"><i><font color="Green" title="E83">A6</font></i></a>, <a class="ref" href="polyform.html#T59" target="_self" title="POLYFORM:th.59">Th59</a></i>;<br/>
<a NAME="E14:97"/>
for <font color="Olive" title="b1">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> &amp; <font color="Olive" title="b1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">n</font> holds <br/><span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:97_1"/><i><font color="Green" title="E87">A10</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span>
 <b>and </b><br/><a NAME="E2:97_1"/><i><font color="Green" title="E88">A11</font></i>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">n</font>
 ;<br/>

<a NAME="E3:97_1"/>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:97_1"><i><font color="Green" title="E87">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:97_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E5:97_1"/>
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="polyform.html#E2:97"><i><font color="Green" title="E79">A2</font></i></a>, <a class="txt" href="polyform.html#E6:97"><i><font color="Green" title="E81">A4</font></i></a>, <a class="txt" href="polyform.html#E7:97"><i><font color="Green" title="E82">A5</font></i></a>, <a class="txt" href="polyform.html#E8:97"><i><font color="Green" title="E83">A6</font></i></a>, <a class="txt" href="polyform.html#E10:97"><i><font color="Green" title="E84">A7</font></i></a>, <a class="txt" href="polyform.html#E2:97_1"><i><font color="Green" title="E88">A11</font></i></a>, <a class="ref" href="polyform.html#T60" target="_self" title="POLYFORM:th.60">Th60</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:97"/>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="bspace.html#K2" title="BSPACE:func.2">Z_2</a> 
 <b>by </b><i><a class="txt" href="polyform.html#E12:97"><i><font color="Green" title="E85">A8</font></i></a>, <a class="txt" href="polyform.html#E13:97"><i><font color="Green" title="E86">A9</font></i></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></i>;<br/>


</div>
