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<div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>thus </b><a NAME="E1:28"/>
( <font color="Maroon">A</font> is <a href="lattice3.html#V1" target="_self">with_suprema</a> implies <font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a>  is <a href="lattice3.html#V2" target="_self">with_infima</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green">E18</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">A</font>  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c</font> &amp; ( for <font color="Olive">c'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c'</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c'</font> holds <br/><font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c'</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" target="_self">LATTICE3:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" target="_self">LATTICE3:def 11</a><br/>

<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E22</font></i>: 
(  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; ( for <font color="Olive">c'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c'</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c'</font> holds <br/><font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">c'</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">z</font> = <font color="Maroon">c</font> <a href="lattice3.html#K8" target="_self">~</a> ;<br/>

<a NAME="E4:28_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c</font> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattice3.html#K8" target="_self">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="lattice3.html#K8" target="_self">~</a>  &amp;  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">y</font> )
 
;<br/>
<b>hence </b><a NAME="E5:28_1"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a>, </i>;<br/>

<b>let </b><font color="Maroon">z'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span>;<br/>

<a NAME="E6:28_1"/><i><font color="Green">E26</font></i>: 
(  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z'</font></span>)</span> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z'</font> )
 
;<br/>
<b>assume </b><a NAME="E7:28_1"/>
( <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E8:28_1"/><b>then </b>
(  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z'</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z'</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:28_1"/><b>then </b>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:28_1"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:28"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span>  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span> st <br/>( <font color="Olive">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> &amp; ( for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span>  st <font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">x</font> &amp; <font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> holds <br/><font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">z</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" target="_self">LATTICE3:def 11</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" target="_self">LATTICE3:def 10</a><br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>;<br/>



<b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span><b> such that </b><br/><a NAME="E4:28"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K8" target="_self">~</a>  &amp; <font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K8" target="_self">~</a>  &amp; ( for <font color="Olive">z'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="lattice3.html#K7" target="_self">~</a> </span>)</span>  st <font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K8" target="_self">~</a>  &amp; <font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K8" target="_self">~</a>  holds <br/><font color="Olive">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">c</font> =  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z</font>;<br/>

<a NAME="E5:28"/><i><font color="Green">E62</font></i>: 
(  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z</font></span>)</span> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K9" target="_self">~</a> <font color="Maroon">z</font> &amp; <font color="Maroon">a</font> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattice3.html#K8" target="_self">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K8" target="_self">~</a>  &amp; <font color="Maroon">b</font> <a href="lattice3.html#K8" target="_self">~</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp;  <a href="lattice3.html#K9" target="_self">~</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K8" target="_self">~</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="lattice3.html#K8" target="_self">~</a>  )
 
;<br/>
<b>hence </b><a NAME="E6:28"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a>, </i>;<br/>

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

<b>assume </b><a NAME="E7:28"/>
( <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font> )
 ;<br/>

<a NAME="E8:28"/><b>then </b>
( <font color="Maroon">c'</font> <a href="lattice3.html#K8" target="_self">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="lattice3.html#K8" target="_self">~</a>  &amp; <font color="Maroon">c'</font> <a href="lattice3.html#K8" target="_self">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> <a href="lattice3.html#K8" target="_self">~</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E9:28"/><b>then </b>
<font color="Maroon">c'</font> <a href="lattice3.html#K8" target="_self">~</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E10:28"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font>
 <b>by </b><i>, </i>;<br/>


</div>
