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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">X</font> is <a href="waybel_6.html#V4" target="_self">order-generating</a> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D5" target="_self">WAYBEL_6:def 5</a><br/>

<b>thus </b><a NAME="E2:34"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E3:34"/><i><font color="Green">E47</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E4:34"/><i><font color="Green">E48</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E50</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/>
<a NAME="E6:34"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:34"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T39">WAYBEL_0:39</a></i>;<br/>
<a NAME="E9:34"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E10:34"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>
<a NAME="E11:34"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:34"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


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