<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>assume </b><a NAME="E1:34"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">L</font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">L</font> is <a href="waybel_2.html#V1">satisfying_MC</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7">WAYBEL_2:def 7</a><br/>

<a NAME="E2:34"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">L</font> is  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM8">LATTICE</a>
 
;<br/>
<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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E2:34_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:34_1"/><b>then </b>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:34"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>  st <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">I</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:34_2"/><i><font color="Green">E42</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><b>assume </b><a NAME="E2:34_2"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E3:34_2"/><b>then </b>
<font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">I</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/><b>then </b><font color="Maroon">y</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_2.html#D6">WAYBEL_2:def 6</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T55">WAYBEL_0:55</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K3">downarrow</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p3">(<span class="default"><span class="p4"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T33">WAYBEL_0:33</a></i>
;<br/><a NAME="E5:34_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:34_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D15">WAYBEL_0:def 15</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font></span>)</span><b> such that </b><br/><a NAME="E9:34_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E12:34_2"/><i><font color="Green">E48</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="yellow_0.html#T38">YELLOW_0:38</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/><a NAME="E13:34_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="xboole_0.html#K1">{}</a> ,<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="E14:34_2"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">i</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><a NAME="E15:34_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/><a NAME="E16:34_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:34_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:34_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">I</font>
 
;<br/>
<a NAME="E3:34_2_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">j</font></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:34_2_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>
<a NAME="E6:34_2_2"/>
<font color="Maroon">y</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">j</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E7:34_2_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:34_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T42">WAYBEL_0:42</a></i>;<br/><b>hence </b><a NAME="E18:34_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D19">WAYBEL_0:def 19</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:34"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>


</div>
