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

<a NAME="E1:12"/>
<span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of 3 : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> 3
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of 3 : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">L</font> being   <a href="subset_1.html#NM2">Subset</a> of 3<b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> &amp; 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">L</font> )
 ;<br/>
<b>thus </b><a NAME="E4:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> 3
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Olive">L</font> where <font color="Olive">L</font> is   <a href="subset_1.html#NM2">Subset</a> of 3 : 2 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Olive">L</font> </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of 3 ;<br/>
<b>take </b>
 <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #)
;<br/>

<a NAME="E3:12"/>
3 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> 3
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<b>hence </b><a NAME="E4:12"/>
(  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pre_topc.html#V1">strict</a> &amp; not  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="struct_0.html#V3">empty</a> &amp; not  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pencil_1.html#V3" target="_self">void</a> &amp; not  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pencil_1.html#V4" target="_self">degenerated</a> &amp; not  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pencil_1.html#V7" target="_self">truly-partial</a> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pencil_1.html#V5" target="_self">with_non_trivial_blocks</a> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pencil_1.html#V6" target="_self">identifying_close_blocks</a> &amp;  <a href="pre_topc.html#G1">TopStruct</a>(# 3,<font color="Maroon">B</font> #) is <a href="pencil_1.html#V8" target="_self">without_isolated_points</a> )
 <b>by </b><i/>;<br/>


</div>
