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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">L</font> =  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <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">T</font></span>)</span>, <font color="Maroon">y</font> be   <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">T</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:44"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:44"/>
 <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/>
<a NAME="E3:44"/><b>then </b>
( <font color="Maroon">x</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="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> &amp;  <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> #) &amp;  <a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span></span>)</span> #) )
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span> ;<br/>
<b>let </b><font color="Maroon">F</font> be  <a href="tex_2.html#V1">proper</a> <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E5:44"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E6:44"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> holds <br/> not <font color="Olive">x</font> <a href="waybel_7.html#R1" target="_self">is_a_cluster_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
 ;<br/>

<b>set </b><font color="Maroon">V</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) </span>}</span> ;<br/>
<a NAME="E7:44"/>
<span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</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">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:44_1"/>
<font color="Maroon">a</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">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">V</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E10:44"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">V</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</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="E1:44_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:44_2"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:44_2"/>
<font color="Maroon">a</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E11:44"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:44_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:44_3"/>
 <a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E3:44_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
;<br/>
<a NAME="E4:44_3"/><b>then </b>
not <font color="Maroon">x</font> <a href="waybel_7.html#R1" target="_self">is_a_cluster_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T4" target="_self">Th4</a>, <a class="txt" href="waybel_7.html#E1:44"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:44_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E7:44_3"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="waybel_7.html#E1:44"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E8:44_3"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:44_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">W</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E13:44"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T5" target="_self">Th5</a>, <a class="ref" href="waybel_3.html#T34">WAYBEL_3:34</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:44_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E2:44_4"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E3:44_4"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> ) )
 ;<br/><b>hence </b><a NAME="E4:44_4"/>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Olive">B</font>] )
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:44"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; ( for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<a NAME="E17:44"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span><b> such that </b><br/><a NAME="E19:44"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">z</font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T2">WAYBEL_0:2</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x'</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font>;<br/>
<a NAME="E21:44"/>
<font color="Maroon">x'</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="waybel_7.html#T8" target="_self">Th8</a>, <a class="ref" href="waybel_0.html#T41">WAYBEL_0:41</a></i>;<br/>
<a NAME="E22:44"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E23:44"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/>
<a NAME="E24:44"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x'</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>;<br/>
<a NAME="E25:44"/><b>then </b><i><font color="Green">E62</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E27:44"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E1:44"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E28:44"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E29:44"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font></span>)</span> ;<br/>
<a NAME="E31:44"/>
( <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_7.html#T8" target="_self">Th8</a>, , , <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>
<a NAME="E32:44"/><b>then </b>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E33:44"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
