<?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#V2">reflexive</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>;<br/>



<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">X</font> is <a href="waybel_0.html#V1" target="_self">directed</a> implies  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> is <a href="waybel_0.html#V1" target="_self">directed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</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">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" target="_self">WAYBEL_0:def 1</a><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#D1" target="_self">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E2:67_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:67_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:67_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:67_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E6:67_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel_0.html#E8:67_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_0.html#E2:67"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E9:67_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 
;<br/>
<b>hence </b><a NAME="E10:67_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E11:67_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E8:67_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel_0.html#E2:67"><i><font color="Green">E35</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>

<b>set </b><font color="Maroon">Y</font> =  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font>;<br/>
<b>assume </b><a NAME="E2:67"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</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>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> holds <br/> ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" target="_self">WAYBEL_0:def 1</a><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#D1" target="_self">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E3:67"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<a NAME="E4:67"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 
;<br/>
<a NAME="E5:67"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:67"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3" target="_self">downarrow</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:67"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">z'</font>
;<br/>

<b>thus </b><a NAME="E10:67"/>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E2:67"><i><font color="Green">E35</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:67"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font> )
 <b>by </b><i>, <a class="txt" href="waybel_0.html#E2:67"><i><font color="Green">E35</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div>
