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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>  <a href="oposet_1.html#L1" title="OPOSET_1:struct.1">OrthoRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:66_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>  <a href="oposet_1.html#L1" title="OPOSET_1:struct.1">OrthoRelStr</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">f</font> = the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <font color="Maroon" title="c2">L'</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">L'</font>,<font color="Maroon" title="c2">L'</font> ;<br/>
<a NAME="E4:66_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L'</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 
<b>by </b><i><a class="ref" href="realset2.html#D5" title="REALSET2:def.5">REALSET2:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:66_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E7:66_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></i>;<br/>
<a NAME="E8:66_1_1"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="oposet_1.html#V2" title="OPOSET_1:attr.2">dneg</a>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="oposet_1.html#T20" title="OPOSET_1:th.20">OPOSET_1:20</a></i>;<br/>
<a NAME="E9:66_1_1"/>
for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>  st <font color="Olive" title="b1">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L'</font>;<br/>

<b>assume </b><a NAME="E1:66_1_1_1"/>
<font color="Maroon" title="c5">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<a NAME="E2:66_1_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></i>;<br/>
<a NAME="E3:66_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E6:66_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></i>;<br/>
<b>hence </b><a NAME="E4:66_1_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="robbins3.html#E2:66_1_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="yellow_0.html#D1" title="YELLOW_0:def.1">YELLOW_0:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:66_1_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D1" title="WAYBEL_9:def.1">WAYBEL_9:def 1</a></i>;<br/>
<a NAME="E11:66_1_1"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="oposet_1.html#V18" title="OPOSET_1:attr.18">Orderinvolutive</a>
 
<b>by </b><i><a class="txt" href="robbins3.html#E8:66_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="oposet_1.html#D32" title="OPOSET_1:def.32">OPOSET_1:def 32</a></i>;<br/>
<a NAME="E12:66_1_1"/>
<font color="Maroon" title="c2">L'</font> is <a href="oposet_1.html#V3" title="OPOSET_1:attr.3">Dneg</a>
 
<b>by </b><i><a class="txt" href="robbins3.html#E8:66_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="oposet_1.html#D6" title="OPOSET_1:def.6">OPOSET_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E13:66_1_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="oposet_1.html#V19" title="OPOSET_1:attr.19">OrderInvolutive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="oposet_1.html#V15" title="OPOSET_1:attr.15">Pure</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="oposet_1.html#V14" title="OPOSET_1:attr.14">PartialOrdered</a> )
 <b>by </b><i><a class="txt" href="robbins3.html#E11:66_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="oposet_1.html#D26" title="OPOSET_1:def.26">OPOSET_1:def 26</a>, <a class="ref" href="oposet_1.html#D33" title="OPOSET_1:def.33">OPOSET_1:def 33</a></i>;<br/>


</div>
