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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V1">with_suprema</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">D1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">D2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:42"/><i><font color="Green">E20</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">D2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T15">WAYBEL_0:15</a></i>;<br/>
<a NAME="E2:42"/><i><font color="Green">E21</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D2</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_0.html#T15">WAYBEL_0:15</a></i>;<br/>
<b>thus </b><a NAME="E3:42"/>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">D2</font></span>)</span>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:42"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">D1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">D2</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:42"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green">E32</font></i>: 
 ex <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">z</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">D2</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:42"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">s</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> <a href="yellow_4.html#K2" target="_self">"\/"</a> <font color="Maroon">D2</font> )
 <b>by </b><i><a class="ref" href="yellow_4.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:42"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E12:42"/>
( <font color="Maroon">D1</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D1</font></span>)</span> &amp; <font color="Maroon">D2</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E13:42"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:42"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D1</font></span>)</span> <a href="yellow_4.html#K2" target="_self">"\/"</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">D2</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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