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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="compts_1.html#V2">compact</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:90"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:90"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:90"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E4:90"/><i><font color="Green">E46</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="90_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:90_1"/><i><font color="Green">E48</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel_9.html#E1:90_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:90_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E1:90_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:90_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E4:90"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/><a NAME="E5:90_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/><a NAME="E6:90_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:90_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_9.html#E5:90_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow_4.html#T37">YELLOW_4:37</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:90"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<a NAME="E7:90"/><i><font color="Green">E49</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <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>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:90_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:90_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:90_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="waybel_9.html#E3:90_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E6:90_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E4:90_2"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:90_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel_9.html#E5:90"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:90"/>
 <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E2:90"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_9.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E9:90"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E10:90"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E3:90"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel_9.html#E7:90"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_9.html#T34" target="_self">Th34</a></i>;<br/>
<b>then 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="E12:90"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E13:90"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E6:90"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>thus </b><a NAME="E14:90"/>
<font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E1:90"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_9.html#E12:90"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>


</div>
