<?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"> ( <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a>  &amp;  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="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  implies <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:57"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V1" title="MATROID0:attr.1">subset-closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:57"/><span class="lab"><font color="Green" title="E26">A2</font></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="finset_1.html#V1" title="FINSET_1:attr.1">finite</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"> <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</a> </span><br/>

<span class="kw">assume </span><a NAME="E3:57"/>
 not <font color="Maroon" title="c1">K</font> is  <a href="simplex0.html#V2" title="SIMPLEX0:attr.2">finite-vertices</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="E4:57"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">K</font> is  <a href="matroid0.html#V3" title="MATROID0:attr.3">finite-membered</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="simplex0.html#T19" target="_self" onmouseover="rs('simplex0/T19')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E5:57"/><span class="kw">then </span>
 not  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font> is  <a href="finset_1.html#V5" title="FINSET_1:attr.5">finite-membered</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:57"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font>
 <span class="kw">and </span><br/><a NAME="E8:57"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 not <font color="Maroon" title="c2">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/>
<a NAME="E9:57"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:57"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<a NAME="E10:57"/><span class="kw">then </span>
 <a href="simplex0.html#K1" title="SIMPLEX0:func.1">subset-closed_closure_of</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="simplex0.html#T3" target="_self" onmouseover="rs('simplex0/T3')" onmouseout="rh()">Th3</a></span>;<br/>
<a NAME="E11:57"/><span class="kw">then </span>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c2">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matroid0.html#K1" title="MATROID0:func.1">the_family_of</a> <font color="Maroon" title="c1">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="simplex0.html#T4" target="_self" onmouseover="rs('simplex0/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:57"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E8:57"><span class="lab"><font color="Green" title="E28">A4</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>
