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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V10">Lattice-like</a> <a href="lattice3.html#V1">with_suprema</a> <a href="lattice3.html#V2">with_infima</a> <a href="oposet_1.html#V14">PartialOrdered</a> <a href="oposet_1.html#V19">OrderInvolutive</a> <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a> <a href="robbins3.html#V11" target="_self">naturally_inf-generated</a>  <a href="robbins3.html#L4" target="_self">OrthoLattRelStr</a> ;<br/>

<a NAME="E1:92"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Olive">y</font></span>)</span> <a href="robbins1.html#K3">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>set </b><font color="Maroon">i</font> = <font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>;<br/>
<a NAME="E1:92_1"/>
( <font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E2:92_1"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">a</font> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a>  &amp; <font color="Maroon">b</font> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:92_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T3">WAYBEL_1:3</a></i>;<br/>
<a NAME="E4:92_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:92_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="lattice3.html#T13">LATTICE3:13</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:92"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Olive">y</font></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <span class="p1">(<span class="default"><font color="Olive">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">s</font> = <font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font>;<br/>
<a NAME="E1:92_2"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E2:92_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="robbins1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> <a href="robbins1.html#K3">`</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:92_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T2">WAYBEL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:92_2"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:92"/>
<font color="Maroon">L</font> is <a href="robbins1.html#V9">de_Morgan</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_3"><b>proof </b></a><div class="add">

<a NAME="E1:92_3"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p2">(<span class="default"><font color="Olive">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="robbins3.html#K4" target="_self">|^|</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">i</font> = <font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font>;<br/>
<a NAME="E1:92_3_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="robbins3.html#K4" target="_self">|^|</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E2:92"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E2:92_3_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="robbins3.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:92_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:92_3_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:92_3_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E5:92_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:92_3_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a>  <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:92_3_1"/><b>then </b>
<font color="Maroon">a</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:92_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="robbins3.html#K4" target="_self">|^|</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="robbins3.html#E2:92_3_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="robbins3.html#E4:92_3_1"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:92_3"/>
<font color="Maroon">L</font> is <a href="robbins1.html#V9">de_Morgan</a>
 <b>by </b><i><a class="ref" href="robbins1.html#D23">ROBBINS1:def 23</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:92"/>
<font color="Maroon">L</font> is <a href="robbins1.html#V9">de_Morgan</a>
 ;<br/>


</div>
