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<div class="add">

<b>thus </b><a NAME="E1:50_1_1"/>
<font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font> is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:50_1_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> )
 ;<br/>

<a NAME="E2:50_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font> is  <a href="yellow_6.html#V2">full</a>  <a href="yellow_6.html#M1">SubNetStr</a> of <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:50_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:50_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E7:50_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E8:50_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E10:50_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E11:50_1_1_1"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font>
 <b>by </b><i/>;<br/>
<a NAME="E12:50_1_1_1"/>
( <font color="Maroon">x1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font> )
 
<b>by </b><i>, , , <a class="ref" href="waybel_9.html#D1" target="_self">Def1</a>, , <a class="ref" href="yellow_6.html#T20">YELLOW_6:20</a></i>;<br/>
<a NAME="E13:50_1_1_1"/><b>then </b>
<font color="Maroon">x1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E14:50_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , , <a class="ref" href="yellow_6.html#T21">YELLOW_6:21</a></i>;<br/>


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

<a NAME="E2:50_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span>  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span> st <br/>( <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span>;<br/>

<b>consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E2:50_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:50_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:50_1_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E6:50_1_1_2"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y1</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:50_1_1_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">y1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>
<a NAME="E9:50_1_1_2"/>
<font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i>, <a class="ref" href="waybel_9.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z1</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E11:50_1_1_2"/>
<font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font> is  <a href="yellow_6.html#V2">full</a>  <a href="yellow_6.html#M1">SubNetStr</a> of <font color="Maroon">N</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:50_1_1_2"/>
( <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="ref" href="waybel_9.html#D1" target="_self">Def1</a>, <a class="ref" href="yellow_6.html#T21">YELLOW_6:21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_1_1"/>
<font color="Maroon">N</font> <a href="waybel_9.html#K5" target="_self">|</a> <font color="Maroon">i</font> is <a href="waybel_0.html#V7">directed</a>
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/>


</div>
