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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">KX</font> being   <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">SX</font> being   <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Olive" title="b1">KX</font> holds   <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b2">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">KX</font> be    <a href="simplex0.html#M1" title="SIMPLEX0:mode.1">SimplicialComplexStr</a> of <font color="Maroon" title="c1">X</font>; <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">SX</font> being   <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KX</font> holds   <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Olive" title="b1">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">SX</font> be    <a href="simplex0.html#M2" title="SIMPLEX0:mode.2">SubSimplicialComplex</a> of <font color="Maroon" title="c2">KX</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1"/>
(  not <font color="Maroon" title="c3">SX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  or <font color="Maroon" title="c3">SX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  or ( not <font color="Maroon" title="c3">SX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c3">SX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
not <font color="Maroon" title="c3">SX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/><div class="add"><a NAME="E2:91_1_1"/>
not <font color="Maroon" title="c3">SX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:91_1_1_1"/>
<font color="Maroon" title="c3">SX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:91_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">SX</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:91_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></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><a NAME="E3:91_1_1"/><span class="kw">then </span>
( not <font color="Maroon" title="c2">KX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  &amp; not <font color="Maroon" title="c2">KX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp;  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_1_1"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_2"/>
<font color="Maroon" title="c3">SX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/><div class="add"><a NAME="E2:91_1_2"/><span class="kw">then </span>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_1_2"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T23" target="_self" title="SIMPLEX0:th.23">Th23</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3"><span class="kw">suppose </span></a><a NAME="E1:91_1_3"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
( not <font color="Maroon" title="c3">SX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  &amp; <font color="Maroon" title="c3">SX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/><div class="add"><a NAME="E2:91_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A3</font></span>: 
not <font color="Maroon" title="c2">KX</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:91_1_3_1"/>
(  not <font color="Maroon" title="c2">KX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  or <font color="Maroon" title="c2">KX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_1_3_1_1"/>
not <font color="Maroon" title="c2">KX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/><div class="add"><a NAME="E2:91_1_3_1_1"/><span class="kw">then </span>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_3"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:91_1_3_1_1"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_1_3_1_2"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c2">KX</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span><br/><div class="add"><a NAME="E2:91_1_3_1_2"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">SX</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D13" target="_self" title="SIMPLEX0:def.13">Def13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">SX</font><span class="kw"> such that </span><br/><a NAME="E4:91_1_3_1_2"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c4">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E5:91_1_3_1_2"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_3"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/><a NAME="E6:91_1_3_1_2"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">SX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91_1_3_1_2"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><a NAME="E7:91_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_3_1_2"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">S1</font> = <font color="Maroon" title="c4">S</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">KX</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1_3_1_2"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E5:91_1_3_1_2"><span class="lab"><font color="Green" title="E42">A7</font></span></a></span>;<br/><a NAME="E9:91_1_3_1_2"/>
<font color="Maroon" title="c5">S1</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_3_1_2"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E6:91_1_3_1_2"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><a NAME="E10:91_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1_3"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:91_1_3_1_2"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E5:91_1_3_1_2"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" title="SIMPLEX0:def.12">Def12</a></span>;<br/><a NAME="E11:91_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T39" title="XXREAL_3:th.39">XXREAL_3:39</a></span>;<br/><a NAME="E12:91_1_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font></span>)</span> <a href="xxreal_3.html#K1" title="XXREAL_3:func.1">+</a> 1</span>)</span> <a href="xxreal_3.html#K3" title="XXREAL_3:func.3">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T24" title="XXREAL_3:th.24">XXREAL_3:24</a></span>;<br/><span class="kw">hence </span><a NAME="E13:91_1_3_1_2"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c3">SX</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c2">KX</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:91_1_3_1_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="xxreal_3.html#T24" title="XXREAL_3:th.24">XXREAL_3:24</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
