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<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>let </b><font color="Maroon">B</font> be    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>;<br/>



<b>consider </b><font color="Maroon">BB</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:47"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">BB</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E3:47"/><i><font color="Green">E42</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K1">UniCl</a> <font color="Maroon">BB</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<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">BT</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/>
<a NAME="E4:47"/>
 <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="E5:47"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <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>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="E6:47"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E7:47"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp; <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> )
 
<b>by </b><i><a class="ref" href="waybel_7.html#T5" target="_self">Th5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">x</font>, <font color="Maroon">Y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E9:47"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font>;<br/><b>assume </b><a NAME="E2:47_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E3:47_1"/>
<font color="Maroon">F</font> is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</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">FF</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/><a NAME="E5:47_1"/>
<font color="Maroon">FF</font> is <a href="tops_2.html#V1">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><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="E1:47_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 ;<br/>

<a NAME="E2:47_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>hence </b><a NAME="E3:47_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:47_1"/>
 ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font> st <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">G</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T8" target="_self">Th8</a>, <a class="ref" href="waybel_3.html#T34">WAYBEL_3:34</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:47"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">F</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font>  st <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">F</font> holds <br/> ex <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Olive">F</font> st <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Olive">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be  <a href="waybel_7.html#V3" target="_self">ultra</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="E1:47_2"/><i><font color="Green">E60</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="E2:47_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  holds <br/>(  not <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> or  not <font color="Olive">p</font> <a href="waybel_7.html#R2" target="_self">is_a_convergence_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font> )
 ;<br/><a NAME="E3:47_2"/>
<font color="Maroon">x</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><a class="ref" href="waybel_7.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E4:47_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</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/><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="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> &amp; not <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> );<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:47_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/><a NAME="E4:47_2_1"/>
not <font color="Maroon">q</font> <a href="waybel_7.html#R2" target="_self">is_a_convergence_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T9" target="_self">Th9</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="E6:47_2_1"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/><a NAME="E7:47_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">AY</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:47_2_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">AY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">AY</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T4" target="_self">Th4</a>, <a class="ref" href="cantor_1.html#D1">CANTOR_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">ay</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:47_2_1"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ay</font> &amp; <font color="Maroon">ay</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AY</font> )
 <b>by </b><i>, <a class="txt" href="waybel_7.html#E3:47"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">ay</font> = <font color="Maroon">ay</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="waybel_7.html#E5:47"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E13:47_2_1"/>
<font color="Maroon">ay</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB</font>
 <b>by </b><i><a class="txt" href="waybel_7.html#E3:47"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_7.html#E5:47"><i><font color="Green">E43</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">BY</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E15:47_2_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">BY</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">BY</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">ay</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">BY</font> )
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><a NAME="E16:47_2_1"/>
<font color="Maroon">ay</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="waybel_7.html#E3:47"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel_7.html#E5:47"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E17:47_2_1"/><b>then </b>
not <font color="Maroon">ay</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/><a NAME="E18:47_2_1"/><b>then </b>
<font color="Maroon">BY</font> is  not   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E20:47_2_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BY</font> &amp; not <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>take </b><font color="Maroon">r</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E21:47_2_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">r</font>] )
 <b>by </b><i><a class="txt" href="waybel_7.html#E5:47"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel_7.html#E9:47"><i><font color="Green">E45</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<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="E7:47_2"/><i><font color="Green">E98</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</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">B</font> )
 <b>and </b><br/><a NAME="E8:47_2"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i><b>reconsider </b><font color="Maroon">FF</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="waybel_7.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E10:47_2"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:47_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:47_2_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="waybel_7.html#T9" target="_self">Th9</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:47_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">FF</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">G</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">FF</font><b> such that </b><br/><a NAME="E12:47_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E13:47_2"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="txt" href="waybel_7.html#E3:47"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/><b>consider </b><font color="Maroon">gg</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">G</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">x</font> = <font color="Maroon">x</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:47_2"/><i><font color="Green">E134</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i><a NAME="E17:47_2"/><i><font color="Green">E135</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_3"><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="E1:47_2_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:47_2_3"/><i><font color="Green">E136</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:47_2_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E5:47_2_3"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<i><font color="Green">E137</font></i>: <font color="Maroon">a</font> = 
<font color="Maroon">x</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_7.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="subset_1.html#K3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">b</font></span>)</span>

;<br/>
<b>consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:47_2_3"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T9" target="_self">Th9</a>, <a class="ref" href="waybel_7.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:47_2_3"/>
( not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> is <a href="waybel_7.html#V2" target="_self">prime</a> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:47_2_3"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:47_2_3"/>
<font color="Maroon">a</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_7.html#T13" target="_self">Th13</a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E18:47_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> is  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T5" target="_self">Th5</a>, <a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then reconsider </b><font color="Maroon">GG</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/><a NAME="E20:47_2"/><i><font color="Green">E139</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">GG</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel_7.html#E9:47"><i><font color="Green">E45</font></i></a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_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="E1:47_2_4"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">GG</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:47_2_4_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:47_2_4_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GG</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:47_2_4_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T13" target="_self">Th13</a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E4:47_2_4_1"/>
not <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#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E3:47_2_4"/><b>then </b>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">z</font> or  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 ;<br/><a NAME="E4:47_2_4"/><b>then </b>
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">G</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">gg</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GG</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">gg</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E5:47"><i><font color="Green">E43</font></i></a>, <a class="txt" href="waybel_7.html#E6:47"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E5:47_2_4"/><b>then </b>
( not <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">x</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">gg</font> )
 <b>by </b><i><a class="txt" href="waybel_7.html#E3:47"><i><font color="Green">E42</font></i></a>, <a class="ref" href="waybel_7.html#T13" target="_self">Th13</a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>hence </b><a NAME="E6:47_2_4"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E22:47_2"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">GG</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E23:47_2"/><b>then </b>
 <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> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T11" target="_self">Th11</a>, <a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/><b>hence </b><a NAME="E24:47_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:47"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="waybel_7.html#E2:47"><i><font color="Green">E33</font></i></a>, </i>;<br/>


</div>
