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

<b>set </b><font color="Maroon">O</font> =  <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>
<b>set </b><font color="Maroon">C</font> = the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>
<b>reconsider </b><font color="Maroon">Emp</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:119"/>
the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  is  <a href="waybel_0.html#V5">antitone</a> <a href="struct_0.html#NM6">Function</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f</font> = the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  as   <a href="struct_0.html#NM6">Function</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  ;<br/>
<a NAME="E2:119_1"/>
for <font color="Olive">x1</font>, <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>   st <font color="Olive">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x2</font> holds <br/>for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>   st <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x1</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font> holds <br/><font color="Olive">y1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> , <font color="Maroon">a2</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ;<br/>

<b>set </b><font color="Maroon">b1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>;<br/>
<a NAME="E1:119_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Emp</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T65">FUNCT_2:65</a></i>;<br/>
<a NAME="E2:119_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T65">FUNCT_2:65</a></i>;<br/>
<b>hence </b><a NAME="E3:119_1_1"/>
( <font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a2</font> implies for <font color="Olive">y1</font>, <font color="Olive">y2</font> being  <a href="struct_0.html#NM1">Element</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>   st <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font> &amp; <font color="Olive">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a2</font> holds <br/><font color="Olive">y1</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y2</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:119_1"/>
the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  is  <a href="waybel_0.html#V5">antitone</a> <a href="struct_0.html#NM6">Function</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D5">WAYBEL_0:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:119"/>
the <a href="robbins1.html#U1">Compl</a> of <a href="oposet_1.html#K3" target="_self">TrivOrthoRelStr</a>  is <a href="oposet_1.html#V18" target="_self">Orderinvolutive</a>
 <b>by </b><i><a class="ref" href="oposet_1.html#T23" target="_self">Th23</a>, </i>;<br/>


</div>
