<?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>assume </b><a NAME="E1:97"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1
 ;<br/>

<b>set </b><font color="Maroon" title="c3">T</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>set </b><font color="Maroon" title="c4">V</font> = <font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font>;<br/>
<a NAME="E2:97"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E80">A2</font></i>: <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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>
<b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="ranknull.html#T9" title="RANKNULL:th.9">RANKNULL:9</a></i>
<br/>.= 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 

;<br/>
<a NAME="E2:97_1"/><i><font color="Green" title="E81">A3</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> 
 
<b>by </b><i><a class="ref" href="polyform.html#T55" target="_self" title="POLYFORM:th.55">Th55</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:97_1"/><i><font color="Green" title="E82">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="txt" href="polyform.html#E2:97_1"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font> <b>by </b><i><a class="txt" href="polyform.html#E4:97_1"><i><font color="Green" title="E82">A4</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">v</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E6:97_1"/><i><font color="Green" title="E83">A5</font></i>: 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="polyform.html#T62" target="_self" title="POLYFORM:th.62">Th62</a></i>;<br/>
<a NAME="E7:97_1"/><i><font color="Green" title="E84">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ranknull.html#T7" title="RANKNULL:th.7">RANKNULL:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">v</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K8" title="POLYFORM:func.8">-polytopes</a> <font color="Maroon" title="c1">p</font></span>)</span> <b>by </b><i><a class="txt" href="polyform.html#E4:97_1"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c8">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span> ;<br/>
<a NAME="E10:97_1"/><i><font color="Green" title="E85">A7</font></i>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E7:97_1"><i><font color="Green" title="E84">A6</font></i></a></i>;<br/>
<a NAME="E11:97_1"/><i><font color="Green" title="E86">A8</font></i>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E1:97_1"><i><font color="Green" title="E80">A2</font></i></a>, <a class="txt" href="polyform.html#E7:97_1"><i><font color="Green" title="E84">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E12:97_1"/><i><font color="Green" title="E87">A9</font></i>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E6:97_1"><i><font color="Green" title="E83">A5</font></i></a>, <a class="txt" href="polyform.html#E10:97_1"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E13:97_1"/><i><font color="Green" title="E88">A10</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E11:97_1"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="polyform.html#E12:97_1"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E14:97_1"/><i><font color="Green" title="E89">A11</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></i>;<br/>
<a NAME="E15:97_1"/><i><font color="Green" title="E90">A12</font></i>: 
2 <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E13:97_1"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="polyform.html#E14:97_1"><i><font color="Green" title="E89">A11</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<i><font color="Green" title="E91">A13</font></i>:  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K22" title="POLYFORM:func.22">-boundary</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="ranknull.html#K2" title="RANKNULL:func.2">.:</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="polyform.html#K17" title="POLYFORM:func.17">-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T45" title="FUNCT_2:th.45">FUNCT_2:45</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="ranknull.html#D2" title="RANKNULL:def.2">RANKNULL:def 2</a></i>
;<br/>
<a NAME="E17:97_1"/><i><font color="Green" title="E92">A14</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <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>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/>
<a NAME="E18:97_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<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></span>)</span>
 
<b>by </b><i><a class="txt" href="polyform.html#E17:97_1"><i><font color="Green" title="E92">A14</font></i></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></i>;<br/>
<a NAME="E19:97_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a> 2
 
<b>by </b><i><a class="txt" href="polyform.html#E1:97"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="polyform.html#T56" target="_self" title="POLYFORM:th.56">Th56</a></i>;<br/>
<b>hence </b><a NAME="E20:97_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="polyform.html#E15:97_1"><i><font color="Green" title="E90">A12</font></i></a>, <a class="txt" href="polyform.html#E16:97_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:97"/>
 <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="polyform.html#K25" title="POLYFORM:func.25">-bounding-chain-space</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="ranknull.html#T6" title="RANKNULL:th.6">RANKNULL:6</a></i>;<br/>


</div>
