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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">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</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>;<br/>







<b>assume </b><a NAME="E1:36"/>
<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"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:36"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">u</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K11">finsups</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">X</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"><font color="Maroon">I</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E5:36"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">u</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/>
<a NAME="E6:36"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><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="E1:36_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:36_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1"/>
<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="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">Z'</font> = <font color="Maroon">Z</font>, <font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ZX</font> = <font color="Maroon">Z'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_2"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_1"><b>suppose </b></a><a NAME="E1:36_2_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_1"/><b>then </b>
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</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>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a>, <a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/><a NAME="E3:36_2_1"/><b>then </b>
( <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><a NAME="E4:36_2_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">i</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><b>take </b>
<font color="Maroon">i</font>
;<br/><b>thus </b><a NAME="E5:36_2_1"/>
( <font color="Maroon">i</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="Maroon">i</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_2_2"><b>suppose </b></a><a NAME="E1:36_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:36_2_2"/><b>then </b><i><font color="Green">E45</font></i>: 
(  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">Z</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Z'</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">ZX</font>,<font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T42">WAYBEL_0:42</a>, <a class="ref" href="yellow_0.html#T54">YELLOW_0:54</a></i>;<br/><a NAME="E3:36_2_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">Z'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T36">YELLOW_0:36</a></i>;<br/><a NAME="E4:36_2_2"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:36_2_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>;<br/><a NAME="E6:36_2_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <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">ZX</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><b>take </b><font color="Maroon">i</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z'</font>;<br/><b>thus </b><a NAME="E7:36_2_2"/>
( <font color="Maroon">i</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="Maroon">i</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a>, <a class="ref" href="waybel_0.html#T42">WAYBEL_0:42</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
