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

<b>set </b><font color="Maroon" title="c4">L</font> =  <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<b>thus </b><a NAME="E1:19_1_1"/>
 <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">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" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">x'</font> = <font color="Maroon" title="c5">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<a NAME="E2:19_1_1_1"/>
<font color="Maroon" title="c6">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">x'</font>
 
;<br/>
<b>hence </b><a NAME="E3:19_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>


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

<b>thus </b><a NAME="E2:19_1_1"/>
 <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">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" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a><br/>

<b>assume </b><a NAME="E1:19_1_1_2"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c10">z'</font> = <font color="Maroon" title="c7">z</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<a NAME="E3:19_1_1_2"/>
( <font color="Maroon" title="c8">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">y'</font> &amp; <font color="Maroon" title="c9">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">z'</font> )
 
<b>by </b><i><a class="txt" href="waybel17.html#E1:19_1_1_2"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<a NAME="E4:19_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">z'</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>


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

<b>thus </b><a NAME="E3:19_1_1"/>
 <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">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="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">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" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:19_1_1_3_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">z'</font> = <font color="Maroon" title="c7">x'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">y'</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c7">x'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">y'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">z</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">z</font>
;<br/>

<a NAME="E5:19_1_1_3_1"/>
( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> &amp; <font color="Maroon" title="c7">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">y'</font> &amp; <font color="Maroon" title="c8">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x'</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c8">y'</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<b>hence </b><a NAME="E6:19_1_1_3_1"/>
( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">z</font> )
 <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1_3"/>
 <a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is <a href="waybel_0.html#V7" title="WAYBEL_0:attr.7">directed</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D6" title="WAYBEL_0:def.6">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" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> is <a href="orders_2.html#V4" title="ORDERS_2:attr.4">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" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel17.html#K3" title="WAYBEL17:func.3">Net-Str</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<b>assume </b><a NAME="E2:19_1_1_4"/>
( <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> )
 ;<br/>

<a NAME="E3:19_1_1_4"/><b>then </b>
( <font color="Maroon" title="c7">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">y'</font> &amp; <font color="Maroon" title="c8">y'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">x'</font> )
 
<b>by </b><i><a class="ref" href="waybel17.html#D3" target="_self" title="WAYBEL17:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1_1_4"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


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


</div>
