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

<b>set </b><font color="Maroon">L</font> =  <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>thus </b><a NAME="E1:19_1_1"/>
 <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="orders_2.html#V2">reflexive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E2:19_1_1_1"/>
<font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font>
 
;<br/>
<b>hence </b><a NAME="E3:19_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/>;<br/>


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

<b>thus </b><a NAME="E2:19_1_1"/>
 <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="orders_2.html#V3">transitive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="E1:19_1_1_2"/><i><font color="Green">E43</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/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font>, <font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E3:19_1_1_2"/>
( <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z'</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:19_1_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:19_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/>;<br/>


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

<b>thus </b><a NAME="E3:19_1_1"/>
 <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="waybel_0.html#V7">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:19_1_1_3"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_3_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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:19_1_1_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">z'</font> = <font color="Maroon">x'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y'</font>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E5:19_1_1_3_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">y'</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<b>hence </b><a NAME="E6:19_1_1_3_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> &amp; <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/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1_3"/>
 <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="waybel_0.html#V7">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>


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

<b>thus </b><a NAME="E4:19_1_1"/>
 <a href="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> is <a href="orders_2.html#V4">antisymmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_4"><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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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="waybel17.html#K3" target="_self">Net-Str</a> <font color="Maroon">a</font>,<font color="Maroon">b</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>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:19_1_1_4"/>
( <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="E3:19_1_1_4"/><b>then </b>
( <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y'</font> &amp; <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:19_1_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


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


</div>
