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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="waybel_8.html#V2">algebraic</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> is <a href="waybel_6.html#V6">co-prime</a> ) )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E3:50"/><i><font color="Green">E33</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green">E34</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green">E36</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="waybel_8.html#V1">satisfying_axiom_K</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_8.html#D4">WAYBEL_8:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as  <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="txt" href="waybel14.html#E1:50"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel14.html#E5:50"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 
;<br/>
<a NAME="E9:50"/>
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>ex <font color="Olive">b<sub>2</sub></font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">b<sub>1</sub></font> st <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="waybel_0.html#V2">filtered</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:50"><i><font color="Green">E32</font></i></a>, <a class="ref" href="waybel14.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E10:50"/><b>then </b>
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="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="struct_0.html#NM2">Subset</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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E8:50"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel14.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E11:50"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T38" target="_self">Th38</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> ;<br/>
<a NAME="E12:50"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><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="E1:50_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
ex <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> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E3:50_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E15:50"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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="E1:50_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</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="E3:50_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>
<a NAME="E4:50_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E5:50_2"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>9</sub></font> is <a href="waybel_6.html#V1">Open</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T2">WAYBEL_8:2</a></i>;<br/>
<a NAME="E6:50_2"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T41">WAYBEL11:41</a></i>;<br/>
<b>hence </b><a NAME="E7:50_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><a NAME="E1:50_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:50_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:50_3_1"/><b>then </b>
ex <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> st <br/>( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> : (  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E4:50_3"/>
<font color="Maroon">c<sub>11</sub></font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>8</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_3_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</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="E1:50_3_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<a NAME="E2:50_3_2_1"/><b>then </b>
ex <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> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E15:50"><i><font color="Green">E39</font></i></a></i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_3_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_3_2_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_3_2_2_1_1"><b>suppose </b></a><a NAME="E1:50_3_2_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:50_3_2_2_1_1"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E3:50_3_2_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_3_2_2_1_2"><b>suppose </b></a><a NAME="E1:50_3_2_2_1_2"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:50_3_2_2_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_3_2_2_1_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:50_3_2_2_1_2_1"/><b>then </b>
ex <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> st <br/>( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:50_3_2_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:50_3_2_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E1:50_3_2_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel14.html#E2:50_3_2_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:50_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E4:50_3_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:50_3_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel14.html#E4:50_3_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:50_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E6:50"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_8.html#D3">WAYBEL_8:def 3</a></i>;<br/>
<a NAME="E8:50_3_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:50_3_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_8.html#K2">compactbelow</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E5:50_3_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E10:50_3_2"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E12:50_3_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="waybel_3.html#V1">compact</a> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:50_3_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_8.html#T4">WAYBEL_8:4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E14:50_3_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E15:50_3_2"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="waybel_6.html#V6">co-prime</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E2:50"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E16:50_3_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E14:50_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/>
<a NAME="E17:50_3_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E19:50_3_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E12:50_3_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="waybel_3.html#D2">WAYBEL_3:def 2</a></i>;<br/>
<a NAME="E20:50_3_2"/>
<font color="Maroon">c<sub>16</sub></font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E22:50_3_2"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:50_3_2_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:50_3_2_3"/>
<font color="Maroon">c<sub>19</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>19</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>18</sub></font><b> such that </b><br/><a NAME="E24:50_3_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel14.html#E19:50_3_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="waybel_3.html#T34">WAYBEL_3:34</a></i>;<br/>
<a NAME="E25:50_3_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>;<br/>
<a NAME="E26:50_3_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel14.html#E24:50_3_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E27:50_3_2"/>
<font color="Maroon">c<sub>19</sub></font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E30:50_3_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E31:50_3_2"/><i><font color="Green">E51</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>20</sub></font>
 <b>and </b><br/><a NAME="E32:50_3_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel14.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>22</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E34:50_3_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E10:50_3_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel14.html#E24:50_3_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel14.html#E31:50_3_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E35:50_3_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E30:50_3_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E36:50_3_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/>
<a NAME="E38:50_3_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>23</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E40:50_3_2"/>
<font color="Maroon">c<sub>23</sub></font> is <a href="waybel_6.html#V6">co-prime</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E15:50_3_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel14.html#E35:50_3_2"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E41:50_3_2"/><b>then </b><i><font color="Green">E55</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> is <a href="waybel_0.html#V2">filtered</a> &amp; <font color="Maroon">c<sub>24</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i><a class="ref" href="waybel14.html#T27" target="_self">Th27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2_4"/><i><font color="Green">E56</font></i>: 
not  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <font color="Maroon">c<sub>24</sub></font> </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_3_2_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E5:50_3_2_4"/><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <font color="Maroon">c<sub>24</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E6:50_3_2_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <font color="Maroon">c<sub>24</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>29</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="E1:50_3_2_4_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <font color="Maroon">c<sub>24</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>30</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="E3:50_3_2_4_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/>
<a NAME="E4:50_3_2_4_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>30</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#T12">WAYBEL11:12</a></i>;<br/>
<b>hence </b><a NAME="E5:50_3_2_4_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> </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> <font color="Maroon">c<sub>24</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="waybel14.html#E5:50_3_2_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/><a NAME="E8:50_3_2_4"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>29</sub></font> is <a href="waybel_0.html#V1">directed</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E41:50_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel14.html#T25" target="_self">Th25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2_4_2"/>
not <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>29</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:50_3_2_4_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font> &amp; not <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>31</sub></font> = <font color="Maroon">c<sub>30</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:50_3_2_4_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E24:50_3_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel14.html#E31:50_3_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel14.html#E3:50_3_2_4_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E7:50_3_2_4_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E30:50_3_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:50_3_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>33</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> ;<br/><a NAME="E10:50_3_2_4_2"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E12:50_3_2_4_2"/>
<font color="Maroon">c<sub>33</sub></font> is <a href="waybel_6.html#V6">co-prime</a>
 <b>by </b><i><a class="txt" href="waybel14.html#E15:50_3_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="waybel14.html#E7:50_3_2_4_2"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E13:50_3_2_4_2"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> is <a href="waybel_0.html#V2">filtered</a> &amp; <font color="Maroon">c<sub>34</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 <b>by </b><i><a class="ref" href="waybel14.html#T27" target="_self">Th27</a></i>;<br/><a NAME="E14:50_3_2_4_2"/><i><font color="Green">E64</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2_4_2_1"/>
not <font color="Maroon">c<sub>31</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>35</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="E3:50_3_2_4_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> &amp; not <font color="Maroon">c<sub>31</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>35</sub></font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><a NAME="E4:50_3_2_4_2_1"/>
<span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2_1"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E5:50_3_2_4_2_1"/><b>then </b>
not <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>35</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:50_3_2_4_2_1"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/><b>hence </b><a NAME="E7:50_3_2_4_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E3:50_3_2_4_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/></div><b>end;</b></div><a NAME="E16:50_3_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/><a NAME="E17:50_3_2_4_2"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="waybel14.html#E13:50_3_2_4_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/><a NAME="E18:50_3_2_4_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>33</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>35</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="E1:50_3_2_4_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>36</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:50_3_2_4_2_2"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>36</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E14:50_3_2_4_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="waybel14.html#E1:50_3_2_4_2_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:50_3_2_4_2_2"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E13:50_3_2_4_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="waybel14.html#E17:50_3_2_4_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:50_3_2_4_2"/>
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_4_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>35</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="E1:50_3_2_4_2_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/>

<a NAME="E2:50_3_2_4_2_3"/>
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:50_3_2_4"><i><font color="Green">E56</font></i></a>, <a class="txt" href="waybel14.html#E6:50_3_2_4_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="waybel14.html#E17:50_3_2_4_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="zfmisc_1.html#T64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E3:50_3_2_4_2_3"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>24</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel14.html#E18:50_3_2_4_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="waybel14.html#E1:50_3_2_4_2_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E20:50_3_2_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E32:50_3_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E10:50_3_2_4"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T22">YELLOW_1:22</a></i>;<br/><a NAME="E11:50_3_2_4"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">c<sub>29</sub></font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E13:50_3_2_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E14:50_3_2_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E19:50_3_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel14.html#E8:50_3_2_4"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_3.html#D1">WAYBEL_3:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>31</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="E16:50_3_2_4"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E13:50_3_2_4"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E17:50_3_2_4"/>
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel14.html#E35:50_3_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E18:50_3_2_4"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2_4"><i><font color="Green">E62</font></i></a></i>;<br/><a NAME="E19:50_3_2_4"/>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E14:50_3_2_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E20:50_3_2_4"/><b>then </b><i><font color="Green">E64</font></i>: 
not <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E16:50_3_2_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="waybel14.html#E18:50_3_2_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/><a NAME="E21:50_3_2_4"/>
<font color="Maroon">c<sub>31</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/><b>hence </b><a NAME="E22:50_3_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E20:50_3_2_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/></div><b>end;</b></div>
<a NAME="E43:50_3_2"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E41:50_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel11.html#T42">WAYBEL11:42</a></i>;<br/>
<a NAME="E44:50_3_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</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="E1:50_3_2_5"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:50_3_2_5"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">c<sub>24</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T33">YELLOW_0:33</a></i>;<br/>
<a NAME="E4:50_3_2_5"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:50_3_2_5"><i><font color="Green">E58</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E5:50_3_2_5"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E45:50_3_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E43:50_3_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E46:50_3_2"/>
<font color="Maroon">c<sub>24</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E3:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E4:50"><i><font color="Green">E34</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E47:50_3_2"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> is <a href="waybel_6.html#V1">Open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E11:50"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel14.html#E41:50_3_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="waybel11.html#T46">WAYBEL11:46</a></i>;<br/>
<a NAME="E48:50_3_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E45:50_3_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="waybel_8.html#T2">WAYBEL_8:2</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>26</sub></font> =  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font></span>)</span>;<br/>

<a NAME="E49:50_3_2"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E48:50_3_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<a NAME="E50:50_3_2"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
;<br/>
<b>hence </b><a NAME="E51:50_3_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="waybel14.html#E44:50_3_2"><i><font color="Green">E57</font></i></a></i>;<br/>

<a NAME="E52:50_3_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E12:50_3_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/>
<a NAME="E53:50_3_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E26:50_3_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel14.html#E31:50_3_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel14.html#E34:50_3_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E54:50_3_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E45:50_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="waybel14.html#E52:50_3_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">c<sub>8</sub></font> ;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font>;<br/><b>thus </b><a NAME="E6:50_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</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="E1:50_3_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<a NAME="E2:50_3_3"/><b>then </b>
ex <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> st <br/>( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Olive">b<sub>1</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E3:50_3_3"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E15:50"><i><font color="Green">E39</font></i></a>, <a class="ref" href="yellow_8.html#T23">YELLOW_8:23</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E18:50"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</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> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> </span>}</span> 
 ;<br/>


</div>
