<?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="robbins3.html#L4" target="_self">OrthoLattRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:95"/>
( <font color="Maroon">L</font> is <a href="robbins3.html#V8" target="_self">involutive</a> &amp; <font color="Maroon">L</font> is <a href="robbins3.html#V9" target="_self">with_Top</a> &amp; <font color="Maroon">L</font> is <a href="robbins1.html#V9">de_Morgan</a> &amp; <font color="Maroon">L</font> is <a href="lattices.html#V10">Lattice-like</a> &amp; <font color="Maroon">L</font> is <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V10">Lattice-like</a> <a href="robbins1.html#V9">de_Morgan</a> <a href="robbins3.html#V8" target="_self">involutive</a> <a href="robbins3.html#V9" target="_self">with_Top</a> <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a>  <a href="robbins3.html#L4" target="_self">OrthoLattRelStr</a>  ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L'</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font> ;<br/>
<a NAME="E4:95"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

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

<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="robbins1.html#D3">ROBBINS1:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="robbins1.html#D3">ROBBINS1:def 3</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:95_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:95"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">f</font> is <a href="oposet_1.html#V2">dneg</a>
 
<b>by </b><i><a class="ref" href="oposet_1.html#D3">OPOSET_1:def 3</a></i>;<br/>
<a NAME="E6:95"/>
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>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:95_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:95_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="robbins1.html#D3">ROBBINS1:def 3</a></i>;<br/>
<a NAME="E3:95_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="robbins1.html#D3">ROBBINS1:def 3</a></i>;<br/>
<a NAME="E4:95_2"/>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a>  = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="robbins3.html#K4" target="_self">|^|</a> <font color="Maroon">y</font></span>)</span> <a href="robbins1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="robbins1.html#D23">ROBBINS1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</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">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E6:95_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="robbins3.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:95"/><b>then </b>
<font color="Maroon">f</font> is <a href="waybel_0.html#V5">antitone</a>
 
<b>by </b><i><a class="ref" href="waybel_9.html#D1">WAYBEL_9:def 1</a></i>;<br/>
<a NAME="E8:95"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">f</font> is <a href="oposet_1.html#V18">Orderinvolutive</a>
 
<b>by </b><i>, <a class="ref" href="oposet_1.html#D32">OPOSET_1:def 32</a></i>;<br/>
<a NAME="E9:95"/>
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds <br/> (  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> <a href="waybel_1.html#R4">is_maximum_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> &amp;  <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> <a href="waybel_1.html#R3">is_minimum_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:95_3"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T20">YELLOW_0:20</a></i>;<br/>

<b>thus </b><a NAME="E2:95_3"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T21">YELLOW_0:21</a></i>;<br/>

<b>set </b><font color="Maroon">t</font> = <font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>;<br/>
<a NAME="E3:95_3"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:95_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 ;<br/>

<font color="Maroon">b</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> = 
<font color="Maroon">b</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <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>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins3.html#K5" target="_self">|_|</a> <font color="Maroon">b</font></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="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<font color="Maroon">b</font> <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="lattices.html#T17">LATTICES:17</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:95_3_1"/><b>then </b>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:95_3_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:95_3"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E5:95_3"/><b>then </b>
<font color="Maroon">L'</font> is <a href="yellow_0.html#V2">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D5">YELLOW_0:def 5</a></i>;<br/>
<a NAME="E6:95_3"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T43">YELLOW_0:43</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> ;<br/>
<b>set </b><font color="Maroon">xx</font> =  <a href="yellow_0.html#K1">"\/"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font>;<br/>
<a NAME="E8:95_3"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">a</font> holds <br/><font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> = 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font>
<b>by </b><i><a class="ref" href="robbins1.html#D3">ROBBINS1:def 3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="robbins3.html#NK4" target="_self">"|_|"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T41">YELLOW_0:41</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p1">(<span class="default"><font color="Maroon">y</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 href="yellow_0.html#K1">"\/"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font>
<b>by </b><i>, <a class="txt" href="robbins3.html#E11:95_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="robbins3.html#E8:95_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E10:95_3"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> <a href="waybel_1.html#R4">is_maximum_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 <b>by </b><i><a class="txt" href="robbins3.html#E11:95_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_1.html#D7">WAYBEL_1:def 7</a></i>;<br/>

<a NAME="E11:95_3"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds  <font color="Olive">a</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3_3"><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/>

<font color="Maroon">a</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> = 
<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"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> 
<b>by </b><i><a class="ref" href="robbins1.html#D23">ROBBINS1:def 23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins3.html#K5" target="_self">|_|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3">`</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font> <a href="robbins3.html#K4" 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="robbins1.html#D23">ROBBINS1:def 23</a></i>
;<br/>
<b>hence </b><a NAME="E2:95_3_3"/>
<font color="Maroon">a</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">t</font> = <font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>;<br/>
<a NAME="E12:95_3"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> holds <br/><font color="Olive">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_3_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:95_3_4"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 ;<br/>

<font color="Maroon">b</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span> = 
<font color="Maroon">b</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3">`</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins3.html#K4" target="_self">|^|</a> <font color="Maroon">b</font></span>)</span> <a href="robbins3.html#K4" 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="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="robbins3.html#K4" 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="lattices.html#T18">LATTICES:18</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E3:95_3_4"/><b>then </b>
<font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<b>hence </b><a NAME="E4:95_3_4"/>
<font color="Maroon">b</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E13:95_3"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> <a href="lattice3.html#R1">is_&lt;=_than</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E14:95_3"/><b>then </b>
<font color="Maroon">L'</font> is <a href="yellow_0.html#V1">lower-bounded</a>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D4">YELLOW_0:def 4</a></i>;<br/>
<a NAME="E15:95_3"/><b>then </b><i><font color="Green">E94</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T42">YELLOW_0:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> ;<br/>
<b>set </b><font color="Maroon">xx</font> =  <a href="yellow_0.html#K2">"/\"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font>;<br/>
<a NAME="E17:95_3"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">a</font> holds <br/><font color="Maroon">t</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> = 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font>
<b>by </b><i><a class="ref" href="robbins1.html#D3">ROBBINS1:def 3</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="robbins3.html#NK3" target="_self">"|^|"</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#T40">YELLOW_0:40</a></i>
<br/>.= 
<font color="Maroon">y</font> <a href="robbins3.html#K4" target="_self">|^|</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="robbins1.html#K3">`</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>,<font color="Maroon">L'</font>
<b>by </b><i>, <a class="ref" href="robbins3.html#T7" target="_self">Th7</a>, <a class="ref" href="robbins3.html#T6" target="_self">Th6</a>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>
;<br/>
<b>hence </b><a NAME="E19:95_3"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K2">}</a></span>,<font color="Maroon">L'</font> <a href="waybel_1.html#R3">is_minimum_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L'</font>
 <b>by </b><i><a class="ref" href="robbins3.html#T7" target="_self">Th7</a>, <a class="ref" href="waybel_1.html#D6">WAYBEL_1:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:95"/><b>then </b>
<font color="Maroon">f</font> <a href="oposet_1.html#R2">OrthoComplement_on</a> <font color="Maroon">L'</font>
 
<b>by </b><i>, <a class="ref" href="oposet_1.html#D36">OPOSET_1:def 36</a></i>;<br/>
<b>hence </b><a NAME="E11:95"/>
( <font color="Maroon">L</font> is <a href="oposet_1.html#V21">Orthocomplemented</a> &amp; <font color="Maroon">L</font> is <a href="oposet_1.html#V14">PartialOrdered</a> )
 <b>by </b><i><a class="ref" href="oposet_1.html#D37">OPOSET_1:def 37</a></i>;<br/>


</div>
