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

<b>set </b><font color="Maroon">N</font> =  <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font>;<br/>
<a NAME="E1:61_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">e</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span> <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:61_1_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">e</font></span><a href="yellow_3.html#K7">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:61_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">e</font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>thus </b><a NAME="E5:61_1_1"/>
 <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font> is <a href="orders_2.html#V2">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_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"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a><br/>

<a NAME="E1:61_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:61_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E6:61_1_1"/>
 <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font> is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_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"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</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="E1:61_1_1_2"/>
( <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="E2:61_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:61_1_1_2"/>
<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="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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

<b>thus </b><a NAME="E7:61_1_1"/>
 <a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font> is <a href="waybel_0.html#V7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_3"><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"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a><br/>

<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E1:61_1_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:61_1_1_3"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel11.html#K4" target="_self">Net-Str</a> <font color="Maroon">e</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a><br/>

<b>assume </b><a NAME="E8:61_1_1"/>
( <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">x</font> )
 ;<br/>

<a NAME="E9:61_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:61_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>


</div>
