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

<span class="kw">let </span><font color="Maroon" title="c1">K</font> be   <a href="simplex0.html#NM1" title="SIMPLEX0:NM.1">SimplicialComplexStr</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 iff <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_1"/>
( <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  or  not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1"/>
<font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 iff <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:69_1_1"/>
(  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 iff <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="69_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_2"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 not <font color="Maroon" title="c1">K</font> is  <a href="pencil_1.html#V3" title="PENCIL_1:attr.3">void</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1 iff <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  )</span><br/><div class="add"><div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a>  implies  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</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>
<div class="add"><span class="kw">assume </span><a NAME="E1:69_1_2_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</a> </span><br/><a NAME="E2:69_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V4" title="MATROID0:attr.4">finite-degree</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_2"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/><span class="kw">assume </span><a NAME="E3:69_1_2_1"/>
<font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#NV9" title="SIMPLEX0:NV.9">with_non-empty_element</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:69_1_2_1"/><span class="kw">then </span>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">K</font> is  <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</a> 
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:69_1_2_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">K</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c3">S</font> = <font color="Maroon" title="c2">S</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1_2_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/><a NAME="E8:69_1_2_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c3">S</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="#E6:69_1_2_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">S</font> = <font color="Maroon" title="c3">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="c1">K</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_2"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:69_1_2_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a></span>;<br/><a NAME="E10:69_1_2_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">S</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</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="#E1:69_1_2"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E1:69_1_2_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E2:69_1_2_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E8:69_1_2_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E11:69_1_2_1"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E12:69_1_2_1"/>
contradiction
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:69_1_2"/><span class="lab"><font color="Green" title="E28">A6</font></span>: 
<font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V4" title="SIMPLEX0:attr.4">empty-membered</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">S</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E5:69_1_2"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="simplex0.html#NV2" title="SIMPLEX0:NV.2">simplex-like</a> 
 <span class="kw">and </span><br/><a NAME="E6:69_1_2"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">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="c1">K</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="#E1:69_1_2"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="simplex0.html#D12" target="_self" onmouseover="rs('simplex0/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E7:69_1_2"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c2">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E8:69_1_2"/>
 <a href="simplex0.html#K6" title="SIMPLEX0:func.6">degree</a> <font color="Maroon" title="c1">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E9:69_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="xxreal_3.html#K2" title="XXREAL_3:func.2">-</a> 1</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="#E6:69_1_2"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="xxreal_3.html#T11" onmouseover="rs('xxreal_3/T11')" onmouseout="rh()">XXREAL_3:11</a></span>;<br/><a NAME="E10:69_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A10</font></span>: 
 not <font color="Maroon" title="c2">S</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ;<br/><a NAME="E11:69_1_2"/>
 the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">K</font> is  <a href="setfam_1.html#V2" title="SETFAM_1:attr.2">empty-membered</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_2"><span class="lab"><font color="Green" title="E28">A6</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:69_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_1_2"><span class="lab"><font color="Green" title="E31">A9</font></span></a>, <a class="txt" href="#E10:69_1_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
