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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:40"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="yellow_3.html#K3" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="yellow_3.html#K3" target="_self">:]</a></span> is <a href="orders_2.html#V3">transitive</a>
 ;<br/>

<a NAME="E2:40"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> holds <br/><font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:40_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E3:40_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E4:40_1"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:40_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">a</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<font color="Maroon">X</font> is <a href="orders_2.html#V3">transitive</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>

<a NAME="E4:40"/>
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font> holds <br/><font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>;<br/>

<b>assume </b><a NAME="E2:40_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E3:40_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
;<br/>
<a NAME="E4:40_2"/><b>then </b>
( <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> &amp; <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">y</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">z</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:40_2"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font></span><a href="yellow_3.html#K7" target="_self">]</a></span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1"><a href="yellow_3.html#K7" target="_self">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">z</font></span><a href="yellow_3.html#K7" target="_self">]</a></span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E6:40_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:40"/>
<font color="Maroon">Y</font> is <a href="orders_2.html#V3">transitive</a>
 <b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div>
