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

<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="polyform.html#NM2" title="POLYFORM:NM.2">polyhedron</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 holds <br/> <a href="vectsp_9.html#K1" title="VECTSP_9:func.1">dim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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</span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 implies  <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 )</span><br/>



<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:99"/><span class="lab"><font color="Green" title="E78">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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</span><br/>

<a NAME="E2:99"/>
 <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="99_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:99_1"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></span>;<br/>
<a NAME="E2:99_1"/><span class="kw">then </span>
 <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="tarski.html#R1" title="TARSKI:pred.1">c=</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E3:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A2</font></span>: 
 <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="tarski.html#R1" title="TARSKI:pred.1">c=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E78">A1</font></span></a>, <a class="ref" href="polyform.html#T56" target="_self" title="POLYFORM:th.56">Th56</a></span>;<br/>
<a NAME="E4:99_1"/>
<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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T55" target="_self" title="POLYFORM:th.55">Th55</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:99_1"/><span class="lab"><font color="Green" title="E80">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_1"><span class="lab"><font color="Green" title="E80">A3</font></span></a></span>;<br/>
<span class="kw">set </span><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="E8:99_1"/><span class="lab"><font color="Green" title="E81">A4</font></span>: 
<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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyform.html#T62" target="_self" title="POLYFORM:th.62">Th62</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:99_1"><span class="lab"><font color="Green" title="E80">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">reconsider </span><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="E11:99_1"/><span class="lab"><font color="Green" title="E82">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T7" title="RANKNULL:th.7">RANKNULL:7</a></span>;<br/>
<a NAME="E12:99_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
;<br/>
<a NAME="E13:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A6</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:99_1"><span class="lab"><font color="Green" title="E81">A4</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E78">A1</font></span></a>, <a class="ref" href="ranknull.html#T9" title="RANKNULL:th.9">RANKNULL:9</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<a NAME="E15:99_1"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:99_1"><span class="lab"><font color="Green" title="E82">A5</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>;<br/>
<a NAME="E16:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A7</font></span>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:99_1"><span class="lab"><font color="Green" title="E83">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></span>;<br/>
<a NAME="E17:99_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T76" title="CARD_2:th.76">CARD_2:76</a></span>;<br/>
<a NAME="E18:99_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A8</font></span>: 
2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:99_1"><span class="lab"><font color="Green" title="E84">A7</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T45" title="FUNCT_2:th.45">FUNCT_2:45</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E78">A1</font></span></a>, <a class="ref" href="ranknull.html#D2" title="RANKNULL:def.2">RANKNULL:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E20:99_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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:99_1"><span class="lab"><font color="Green" title="E85">A8</font></span></a>, <a class="txt" href="#E3:99_1"><span class="lab"><font color="Green" title="E79">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:99"/>
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ranknull.html#T6" title="RANKNULL:th.6">RANKNULL:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
