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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V2">with_infima</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:112"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K12" target="_self">fininfs</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="waybel_0.html#K12" target="_self">fininfs</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" target="_self">fininfs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16" target="_self">Th16</a>, <a class="ref" href="waybel_0.html#T50" target="_self">Th50</a></i>;<br/>
<b>hence </b><a NAME="E2:112"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" target="_self">fininfs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E3:112"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></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="E4:112"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4" target="_self">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" target="_self">fininfs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:112"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12" target="_self">fininfs</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E4:112"><i><font color="Green">E53</font></i></a>, <a class="ref" href="waybel_0.html#D16" target="_self">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:112"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="waybel_0.html#E7:112"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E12:112"/><i><font color="Green">E56</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:112_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:112_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="waybel_0.html#E12:112"><i><font color="Green">E56</font></i></a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/><b>hence </b><a NAME="E3:112_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E12:112"><i><font color="Green">E56</font></i></a>, <a class="ref" href="waybel_0.html#D20" target="_self">Def20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:112"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="txt" href="waybel_0.html#E3:112"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E15:112"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_0.html#E9:112"><i><font color="Green">E55</font></i></a>, <a class="txt" href="waybel_0.html#E13:112"><i><font color="Green">E57</font></i></a>, <a class="ref" href="waybel_0.html#T43" target="_self">Th43</a></i>;<br/>
<b>hence </b><a NAME="E16:112"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel_0.html#E7:112"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel_0.html#D20" target="_self">Def20</a></i>;<br/>


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