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<div class="add">

<b>let </b><font color="Maroon">A</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> st <br/>( <font color="Olive">X</font> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Olive">a</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">X</font> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D12" target="_self">LATTICE3:def 12</a><br/>

<b>thus </b><a NAME="E2:32"/>
<font color="Maroon">A</font> is <a href="lattice3.html#V1" target="_self">with_suprema</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_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">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">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:32_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</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 <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</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">c</font>
;<br/>

<a NAME="E3:32_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:32_1"/>
( <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>, </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="E5:32_1"/><i><font color="Green">E25</font></i>: 
( <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="E6:32_1"/>
<span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Maroon">c'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</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#D9" target="_self">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K7">}</a></span>
 ;<br/>

<b>hence </b><a NAME="E2:32_1_1"/>
<font color="Maroon">d</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:32_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c'</font>
 <b>by </b><i/>;<br/>


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

<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#D11" target="_self">LATTICE3:def 11</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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> ;<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="E4:32"/><i><font color="Green">E26</font></i>: 
( <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="lattice3.html#R2" target="_self">is_&lt;=_than</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 <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="lattice3.html#R2" target="_self">is_&lt;=_than</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">c</font>
;<br/>

<a NAME="E5:32"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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#D9" target="_self">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:32_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:32_2"/><b>then </b>
 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="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c'</font> &amp; <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_2"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>

<a NAME="E7:32"/>
<span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span>  <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</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#D9" target="_self">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:32_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:32_3"/><b>then </b>
 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="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">c'</font> &amp; <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 
;<br/>
<b>hence </b><a NAME="E3:32_3"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:32"/>
<font color="Maroon">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><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="E9:32"/>
( <font color="Maroon">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E10:32"/><b>then </b>
<font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">c</font> where <font color="Olive">c</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> : ( <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Olive">c</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">b</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E11:32"/>
<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>
