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

<b>let </b><font color="Maroon" title="c1">p</font> be    <a href="polyform.html#M1" title="POLYFORM:mode.1">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">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="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <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="c5">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="c6">m</font>, <font color="Maroon" title="c7">n</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="E1:94"/><i><font color="Green" title="E76">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="E2:94"/><i><font color="Green" title="E77">A2</font></i>: 
<font color="Maroon" title="c5">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="E3:94"/><i><font color="Green" title="E78">A3</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="E4:94"/><i><font color="Green" title="E79">A4</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 <b>and </b><br/><a NAME="E5:94"/><i><font color="Green" title="E80">A5</font></i>: 
<font color="Maroon" title="c6">m</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>and </b><br/><a NAME="E6:94"/><i><font color="Green" title="E81">A6</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="E7:94"/><i><font color="Green" title="E82">A7</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>and </b><br/><a NAME="E8:94"/><i><font color="Green" title="E83">A8</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">n</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">iseq</font> =  <a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">v</font>;<br/>
<a NAME="E9:94"/>
(  <a href="real_1.html#K1" title="REAL_1:func.1">-</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#E1:94"><i><font color="Green" title="E76">A1</font></i></a></i>;<br/>
<a NAME="E10:94"/><b>then </b><i><font color="Green" title="E84">A9</font></i>: 
<font color="Maroon" title="c6">m</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> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="polyform.html#E3:94"><i><font color="Green" title="E78">A3</font></i></a>, <a class="txt" href="polyform.html#E4:94"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="polyform.html#E5:94"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="polyform.html#E6:94"><i><font color="Green" title="E81">A6</font></i></a>, <a class="txt" href="polyform.html#E7:94"><i><font color="Green" title="E82">A7</font></i></a>, <a class="txt" href="polyform.html#E8:94"><i><font color="Green" title="E83">A8</font></i></a>, <a class="ref" href="polyform.html#T35" target="_self" title="POLYFORM:th.35">Th35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:94_1"/>
<font color="Maroon" title="c5">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</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></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> 
 ;<br/><a NAME="E2:94_1"/><b>then </b>
<font color="Maroon" title="c6">m</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>by </b><i><a class="txt" href="polyform.html#E2:94"><i><font color="Green" title="E77">A2</font></i></a>, <a class="ref" href="bspace.html#T9" title="BSPACE:th.9">BSPACE:9</a></i>;<br/><b>hence </b><a NAME="E3:94_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polyform.html#E10:94"><i><font color="Green" title="E84">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E12:94"/><b>then </b><i><font color="Green" title="E85">A10</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="bspace.html#K3" title="BSPACE:func.3">@</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</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></span>)</span> <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="ref" href="bspace.html#T11" title="BSPACE:th.11">BSPACE:11</a></i>;<br/>
<a NAME="E13:94"/><i><font color="Green" title="E86">A11</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</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#E1:94"><i><font color="Green" title="E76">A1</font></i></a>, <a class="ref" href="polyform.html#D3" target="_self" title="POLYFORM:def.3">Def3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">m</font> = 
<span class="p1">(<span class="default"><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> </span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polyform.html#K16" title="POLYFORM:func.16">incidence-value</a> <font color="Maroon" title="c4">e</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">m</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></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyform.html#E4:94"><i><font color="Green" title="E79">A4</font></i></a>, <a class="txt" href="polyform.html#E5:94"><i><font color="Green" title="E80">A5</font></i></a>, <a class="txt" href="polyform.html#E12:94"><i><font color="Green" title="E85">A10</font></i></a>, <a class="txt" href="polyform.html#E13:94"><i><font color="Green" title="E86">A11</font></i></a>, <a class="ref" href="polyform.html#D14" target="_self" title="POLYFORM:def.14">Def14</a></i>
<br/>.= 
 <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="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
;<br/>
<b>hence </b><a NAME="E15:94"/>
<span class="p1">(<span class="default"><a href="polyform.html#K19" title="POLYFORM:func.19">incidence-sequence</a> <font color="Maroon" title="c4">e</font>,<font color="Maroon" title="c5">v</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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> 
 ;<br/>


</div>
