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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E20</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">OT</font> =  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:14"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">Z</font>, <font color="Olive">Z'</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">OT</font>  holds not <font color="Olive">Z</font> <a href="subset_1.html#R1">misses</a> <font color="Olive">Z'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">OT</font>, <font color="Maroon">Z'</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">OT</font>;<br/>

<a NAME="E1:14_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">Z'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<b>hence </b><a NAME="E2:14_1"/>
not <font color="Maroon">Z</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Z'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:14"/>
not  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="orders_3.html#V2" target="_self">disconnected</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="orders_3.html#V2" target="_self">disconnected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:14_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> &amp; the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">A</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">B</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">B'</font> = <font color="Maroon">B</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">OT</font> <b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E5:14_2"/>
<font color="Maroon">A'</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">B'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:14_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:14"/>
not <font color="Maroon">T</font> is <a href="orders_3.html#V3" target="_self">disconnected</a>
 <b>by </b><i/>;<br/>


</div>
