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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">F</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:105"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E2:105"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> )
 <b>and </b><br/><a NAME="E3:105"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">Y</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> <a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2" target="_self">WAYBEL_0:def 2</a><br/>

<b>assume </b><a NAME="E4:105"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y1</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:105"/><i><font color="Green">E34</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y1</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y1</font>,<font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T5" target="_self">Th5</a></i>;<br/>
<b>assume </b><a NAME="E7:105"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y2</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:105"/><i><font color="Green">E35</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y2</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y2</font>,<font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T5" target="_self">Th5</a></i>;<br/>
<b>take </b><font color="Maroon">z</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font></span>)</span>,<font color="Maroon">L</font>;<br/>

<a NAME="E10:105"/><i><font color="Green">E37</font></i>: 
( ( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or <font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<b>hence </b><a NAME="E11:105"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D3" target="_self">Def3</a>, <a class="ref" href="waybel_0.html#D4" target="_self">Def4</a></i>;<br/>

<a NAME="E12:105"/>
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> &amp; <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y2</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D4" target="_self">Def4</a>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E13:105"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D4" target="_self">Def4</a>, , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>


</div>
