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

<b>set </b><font color="Maroon" title="c1">O</font> =  <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ;<br/>
<b>set </b><font color="Maroon" title="c2">C</font> = the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c3">Emp</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  <b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:118"/>
the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  is  <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c4">f</font> = the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  ;<br/>
<a NAME="E2:118_1"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>   st <font color="Olive" title="b1">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">x2</font> holds <br/>for <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>   st <font color="Olive" title="b3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x1</font> &amp; <font color="Olive" title="b4">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b3">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b4">y2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a1</font>, <font color="Maroon" title="c6">a2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a2</font> implies for <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>   st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a1</font> &amp; <font color="Olive" title="b2">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a2</font> holds <br/><font color="Olive" title="b1">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">y2</font> )</span><br/>

<b>set </b><font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a1</font>;<br/>
<a NAME="E1:118_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Emp</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="funct_2.html#T65" title="FUNCT_2:th.65">FUNCT_2:65</a></i>;<br/>
<a NAME="E2:118_1_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a1</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</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="E3:118_1_1"/>
( <font color="Maroon" title="c5">a1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a2</font> implies for <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>   st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a1</font> &amp; <font color="Olive" title="b2">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a2</font> holds <br/><font color="Olive" title="b1">y1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">y2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:118_1"/>
the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  is  <a href="waybel_0.html#V5" title="WAYBEL_0:attr.5">antitone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> ,<a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D5" title="WAYBEL_0:def.5">WAYBEL_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:118"/>
the <a href="robbins1.html#U1" title="ROBBINS1:sel.1">Compl</a> of <a href="oposet_1.html#K3" title="OPOSET_1:func.3">TrivOrthoRelStr</a>  is <a href="oposet_1.html#V17" title="OPOSET_1:attr.17">Orderinvolutive</a>
 <b>by </b><i><a class="ref" href="oposet_1.html#D32" target="_self" title="OPOSET_1:def.32">Def32</a>, <a class="ref" href="oposet_1.html#T19" target="_self" title="OPOSET_1:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
