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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel33.html#V1" target="_self">lim-inf</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:32"/><i><font color="Green">E46</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="waybel28.html#K4">xi</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:32"/>
 <a href="waybel28.html#K4">xi</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel28.html#D4">WAYBEL28:def 4</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="waybel28.html#K4">xi</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">V</font> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Olive">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_6.html#D27">YELLOW_6:def 27</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:32_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E2:32_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="waybel12.html#D1">WAYBEL12:def 1</a></i>;<br/><a NAME="E3:32_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K4">xi</a> <font color="Maroon">L</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">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:32_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E6:32_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/>for <font color="Olive">N</font> being  <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">N</font>,<font color="Olive">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L</font> holds <br/><font color="Olive">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">F</font> be  <a href="waybel_7.html#V3">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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/><a NAME="E7:32_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of  <a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font> holds   <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font>
 <b>by </b><i/>;<br/><a NAME="E8:32_1"/>
(  <a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font> &amp;  <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><a NAME="E9:32_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel28.html#D3">WAYBEL28:def 3</a></i>;<br/><b>assume </b><a NAME="E10:32_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>assume </b><a NAME="E11:32_1"/>
not  <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E12:32_1"/><b>then </b>
 <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/><a NAME="E13:32_1"/><b>then </b>
 <a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E15:32_1"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>  st <font color="Maroon">i</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/><a NAME="E16:32_1"/><i><font color="Green">E124</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">f</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> : <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/><a NAME="E17:32_1"/>
<font color="Maroon">i</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="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">f</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E19:32_1"/><i><font color="Green">E126</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font>, <font color="Maroon">f'</font> = <font color="Maroon">f</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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">B</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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E22:32_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">A'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">f'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">B</font>;<br/><a NAME="E24:32_1"/>
not  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></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#T8">WAYBEL_7:8</a></i>;<br/><a NAME="E25:32_1"/><b>then </b>
not <font color="Maroon">B</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/><a NAME="E26:32_1"/><b>then </b><i><font color="Green">E128</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E27:32_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/><a NAME="E28:32_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">B</font></span><a href="tarski.html#K4">]</a></span> <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="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">B</font></span><a href="tarski.html#K4">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> ;<br/><a NAME="E30:32_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/><a NAME="E31:32_1"/><b>then </b>
<font color="Maroon">j</font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E32:32_1"/><b>then </b>
<font color="Maroon">j</font> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">i</font> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><a NAME="E33:32_1"/><b>then </b>
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/><a NAME="E34:32_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i/>;<br/><a NAME="E35:32_1"/><b>then </b>
<font color="Maroon">j</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/><a NAME="E36:32_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/><b>hence </b><a NAME="E37:32_1"/>
contradiction
 <b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:32"/><i><font color="Green">E130</font></i>: 
for <font color="Olive">F</font> being <a href="waybel_7.html#V3">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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>  st <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/> <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Olive">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>set </b><font color="Maroon">V</font> = <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:32_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>let </b><font color="Maroon">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E3:32_2"/><i><font color="Green">E132</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">N</font>,<font color="Maroon">p</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L</font>
 ;<br/><a NAME="E4:32_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="subset_1.html#D5">SUBSET_1:def 5</a></i>;<br/><a NAME="E5:32_2"/><b>then </b><i><font color="Green">E133</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">A</font>
 ;<br/><b>assume </b><a NAME="E6:32_2"/>
not <font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 ;<br/><a NAME="E7:32_2"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T10">WAYBEL_0:10</a></i>;<br/><b>then consider </b><font color="Maroon">N'</font> being  <a href="waybel_0.html#V6">strict</a>  <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E9:32_2"/><i><font color="Green">E134</font></i>: 
(  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">N'</font> is    <a href="yellow_6.html#M1">SubNetStr</a> of <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/><a NAME="E10:32_2"/>
 <a href="waybel28.html#K3">lim_inf-Convergence</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="yellow_6.html#D21">YELLOW_6:def 21</a></i>;<br/><a NAME="E11:32_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:32_2"/><i><font color="Green">E136</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#T19">YELLOW_6:19</a></i>;<br/><b>consider </b><font color="Maroon">N1</font> being  <a href="waybel_0.html#V6">strict</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:32_2"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">N1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E15:32_2"/><i><font color="Green">E156</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/><a NAME="E16:32_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K2">the_universe_of</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, , , <a class="ref" href="classes1.html#D1">CLASSES1:def 1</a></i>;<br/><a NAME="E17:32_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">N'</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D14">YELLOW_6:def 14</a></i>;<br/><a NAME="E18:32_2"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N'</font>
 <b>by </b><i>, , <a class="ref" href="waybel28.html#D3">WAYBEL28:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">M</font> be    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N'</font>;<br/><a NAME="E1:32_2_1"/>
<font color="Maroon">M</font> is    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T24">YELLOW_6:24</a></i>;<br/><b>hence </b><a NAME="E2:32_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="waybel28.html#D3">WAYBEL28:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E20:32_2"/><b>then </b><i><font color="Green">E164</font></i>: 
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N'</font>  st <font color="Olive">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Olive">M</font>
 ;<br/><b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span> ;<br/><b>consider </b><font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font>;<br/><b>set </b><font color="Maroon">g</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j2</font></span>)</span>;<br/><a NAME="E22:32_2"/><i><font color="Green">E165</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span> 
 ;<br/><a NAME="E23:32_2"/>
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span>  holds <br/><font color="Olive">g</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#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E3:32_2_2"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j3</font></span>)</span>
 ;<br/>
<a NAME="E4:32_2_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j3</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/>
<a NAME="E5:32_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E6:32_2_2"/><b>then </b><i><font color="Green">E168</font></i>: 
<font color="Maroon">g</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:32_2_2"/>
<font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:32_2_2"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E9:32_2_2"/>
<font color="Maroon">g</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#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </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#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E25:32_2"/><i><font color="Green">E169</font></i>: 
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_2_3"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:32_2_3"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <font color="Olive">Y</font>,<span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p4">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">Y</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">G</font> :  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Olive">Y</font>, <a href="yellow_1.html#K3">BoolePoset</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span> </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">Y</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:32_2_3"/><i><font color="Green">E170</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>, <a href="yellow_1.html#K3">BoolePoset</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span> )
 ;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<i><font color="Green">E171</font></i>:  <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> = 
 <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D12">YELLOW_0:def 12</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T19">YELLOW_1:19</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> ex <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> st <br/>( <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span> );<br/>
<a NAME="E7:32_2_3"/><i><font color="Green">E173</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_3_2"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E2:32_2_3_2"/><i><font color="Green">E175</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font> holds <br/> not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>

<a NAME="E3:32_2_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E5:32_2_3_2"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span>
 ;<br/>
<b>thus </b><a NAME="E6:32_2_3_2"/>
contradiction
 <b>by </b><i>, </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="E9:32_2_3"/><i><font color="Green">E177</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> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font> )
 <b>and </b><br/><a NAME="E10:32_2_3"/><i><font color="Green">E178</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</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">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N'</font></span>)</span> <b>by </b><i>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/>
<a NAME="E12:32_2_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N'</font> is <a href="waybel_0.html#V1">directed</a>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D6">WAYBEL_0:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon">j0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E14:32_2_3"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">j0</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">N'</font> &amp; <font color="Maroon">j0</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T1">WAYBEL_0:1</a></i>;<br/>
<a NAME="E15:32_2_3"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_3_3"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:32_2_3_3"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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#T12">FUNCT_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> <b>by </b><i/>;<br/>
<a NAME="E4:32_2_3_3"/><i><font color="Green">E183</font></i>: 
<font color="Maroon">f1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j0</font>
 
<b>by </b><i>, <a class="txt" href="waybel33.html#E1:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E6:32_2_3_3"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j1</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:32_2_3_3"/>
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>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> holds <br/><font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_3_1"><b>proof </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="E1:32_2_3_3_1"/><i><font color="Green">E185</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span>
 ;<br/>

<a NAME="E2:32_2_3_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:32_2_3_3_1"/><i><font color="Green">E187</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:32_2_3_3_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E6:32_2_3_3_1"/><b>then </b><i><font color="Green">E188</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:32_2_3_3_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">n'</font> where <font color="Olive">n'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">j0</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">n'</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel_9.html#T12">WAYBEL_9:12</a></i>;<br/>
<b>then consider </b><font color="Maroon">n'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E9:32_2_3_3_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">j0</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">n'</font> )
 ;<br/>
<a NAME="E10:32_2_3_3_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E11:32_2_3_3_1"/>
<font color="Maroon">f1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">n'</font>
 
<b>by </b><i><a class="txt" href="waybel33.html#E3:32"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>
<a NAME="E12:32_2_3_3_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">m'</font> where <font color="Olive">m'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">j1</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">m'</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:32_2_3_3_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_9.html#T12">WAYBEL_9:12</a></i>;<br/>
<a NAME="E14:32_2_3_3_1"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E15:32_2_3_3_1"/><b>then </b><i><font color="Green">E193</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E16:32_2_3_3_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E17:32_2_3_3_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E18:32_2_3_3_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_2_3_3"/>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:32_2_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<a NAME="E17:32_2_3"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="yellow_1.html#T20">YELLOW_1:20</a></i>;<br/>
<b>hence </b><a NAME="E18:32_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:32_2"/>
for <font color="Olive">y</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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>  st  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">y</font> holds <br/> not <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_4"><b>proof </b></a><div class="add">

<b>let </b><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#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:32_2_4"/><i><font color="Green">E194</font></i>: 
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:32_2_4"/><i><font color="Green">E195</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/>
<a NAME="E3:32_2_4"/>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:32_2_4"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:32_2"/><b>then </b>
not  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/><a NAME="E28:32_2"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font></span>)</span> is <a href="tex_2.html#V1">proper</a>
 <b>by </b><i><a class="ref" href="waybel_7.html#T8">WAYBEL_7:8</a></i>;<br/><b>then consider </b><font color="Maroon">F</font> being   <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E30:32_2"/><i><font color="Green">E196</font></i>: 
(  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> is <a href="waybel_7.html#V3">ultra</a> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T30">WAYBEL_7:30</a></i>;<br/><b>consider </b><font color="Maroon">j0</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font>;<br/><a NAME="E32:32_2"/><i><font color="Green">E197</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>reconsider </b><font color="Maroon">rj</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span>, <font color="Maroon">a</font> = <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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/><a NAME="E34:32_2"/><i><font color="Green">E199</font></i>: 
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font> &amp;  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a>, <a class="ref" href="waybel_0.html#T50">WAYBEL_0:50</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_5"><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="E1:32_2_5"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rj</font>
 ;<br/><a NAME="E2:32_2_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#D7">WAYBEL_9:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">q</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:32_2_5"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:32_2_5"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font></span>)</span> <a href="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E6:32_2_5"/><b>then </b><i><font color="Green">E202</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j0</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:32_2_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E8:32_2_5"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/></div><b>end;</b></div><a NAME="E36:32_2"/><b>then </b>
<font color="Maroon">rj</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E37:32_2"/><b>then </b>
<font color="Maroon">rj</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E38:32_2"/><b>then </b>
<font color="Maroon">rj</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/><a NAME="E39:32_2"/><b>then </b><i><font color="Green">E203</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/><a NAME="E40:32_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">x</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="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E41:32_2"/><i><font color="Green">E205</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</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">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span> ,<font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E42:32_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</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">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_6"><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="E1:32_2_6"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : verum </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">j3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E3:32_2_6"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j3</font></span>)</span>
 <b>by </b><i/>;<br/>
<i><font color="Green">E208</font></i>: <font color="Maroon">x</font> = 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="txt" href="waybel33.html#E1:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_9.html#D2">WAYBEL_9:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j3</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">f1</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j3</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E6:32_2_6"/>
 <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:32_2_6"/><b>then </b>
( <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel33.html#E3:32"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:32_2"/><b>then </b><i><font color="Green">E209</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/><a NAME="E44:32_2"/>
<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">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:32_2_7"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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">f</font></span>)</span> where <font color="Olive">f</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">f0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:32_2_7"/><i><font color="Green">E211</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">f0</font> &amp; <font color="Maroon">f0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Maroon">N'</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">f0</font> );<br/>
<div><i><font color="Green">E212</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font>;<br/><a NAME="E1:32_2_7_1"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E2:32_2_7_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp;  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:32_2_7_1"/><b>then </b><i><font color="Green">E213</font></i>: 
<font color="Maroon">f0</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E1:32"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel_7.html#T13">WAYBEL_7:13</a></i>;<br/><a NAME="E4:32_2_7_1"/>
<font color="Maroon">F</font> is  <a href="waybel_7.html#V3">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> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:32_2_7_1"/><b>then </b>
not  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span></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#T8">WAYBEL_7:8</a></i>;<br/><a NAME="E6:32_2_7_1"/><b>then </b>
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/><b>then consider </b><font color="Maroon">nj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:32_2_7_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">nj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f0</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><a NAME="E9:32_2_7_1"/>
( <font color="Maroon">nj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f0</font> &amp; <font color="Maroon">nj</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> )
 <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">pj'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:32_2_7_1"/><i><font color="Green">E217</font></i>: 
( <font color="Maroon">pj'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">nj</font> <a href="hidden.html#R1">=</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pj'</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:32_2_7_1"/>
<font color="Maroon">pj'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E13:32_2_7_1"/><b>then </b>
<font color="Maroon">pj'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">y'</font> where <font color="Olive">y'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> : <font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y'</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_9.html#T12">WAYBEL_9:12</a></i>;<br/><b>then consider </b><font color="Maroon">pj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font><b> such that </b><br/><a NAME="E15:32_2_7_1"/><i><font color="Green">E219</font></i>: 
( <font color="Maroon">pj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pj'</font> &amp; <font color="Maroon">j</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">pj</font> )
 ;<br/><b>reconsider </b><font color="Maroon">pj'</font> = <font color="Maroon">pj'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <b>by </b><i/>;<br/><i><font color="Green">E220</font></i>: <font color="Maroon">N'</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">pj</font> = 
<span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">pj'</font>
<b>by </b><i>, <a class="ref" href="waybel_9.html#T16">WAYBEL_9:16</a></i>
<br/>.= 
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">j</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">pj'</font>

;<br/><b>take </b><font color="Maroon">pj</font> = <font color="Maroon">pj</font>;<br/><b>thus </b><a NAME="E18:32_2_7_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">pj</font>]
 <b>by </b><i>, , , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N'</font>,<font color="Maroon">N'</font><b> such that </b><br/><a NAME="E6:32_2_7"/><i><font color="Green">E221</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>,<font color="Maroon">p</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S10">FUNCT_2:sch 10</a>(<a class="txt" href="waybel33.html#E3:32"><i><font color="Green">E47</font></i></a>);<br/></i>
<a NAME="E7:32_2_7"/><i><font color="Green">E222</font></i>: 
for <font color="Olive">M</font> being   <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N'</font>  st <font color="Olive">M</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K7">NetUniv</a> <font color="Maroon">L</font> holds <br/><font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <font color="Olive">M</font>
 
<b>by </b><i>, , <a class="ref" href="waybel28.html#T3">WAYBEL28:3</a></i>;<br/>
<a NAME="E8:32_2_7"/>
<font color="Maroon">p</font> is <a href="waybel28.html#V1">greater_or_equal_to_id</a>
 
<b>by </b><i>, <a class="ref" href="waybel28.html#D1">WAYBEL28:def 1</a></i>;<br/>
<a NAME="E9:32_2_7"/><b>then </b><i><font color="Green">E223</font></i>: 
 <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel28.html#T14">WAYBEL28:14</a></i>;<br/>
<a NAME="E10:32_2_7"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> holds <br/><font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_7_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span>
 ;<br/>

<a NAME="E2:32_2_7_2"/><b>then </b>
<font color="Maroon">b</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"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="waybel11.html#T19">WAYBEL11:19</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span><b> such that </b><br/><a NAME="E4:32_2_7_2"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 ;<br/>
<a NAME="E5:32_2_7_2"/><i><font color="Green">E225</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#T7">WAYBEL28:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N'</font> <b>by </b><i><a class="ref" href="waybel28.html#T7">WAYBEL28:7</a></i>;<br/>
<a NAME="E7:32_2_7_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:32_2_7_2"/><b>then </b><i><font color="Green">E226</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="waybel28.html#D2">WAYBEL28:def 2</a></i>
<br/>.= 
<font color="Maroon">N'</font> <a href="waybel_0.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E10:32_2_7_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f0</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:32_2_7"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:32_2_7"/><b>then </b><i><font color="Green">E227</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">f0</font>,<font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>
 <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="ref" href="waybel_9.html#D2">WAYBEL_9:def 2</a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><font color="Maroon">N'</font> <a href="waybel28.html#K1">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E14:32_2_7"/>
<font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="waybel33.html#E1:32"><i><font color="Green">E46</font></i></a>, , , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E45:32_2"/><b>then </b>
 <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E46:32_2"/><b>then </b><i><font color="Green">E228</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/><a NAME="E47:32_2"/>
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/><b>hence </b><a NAME="E48:32_2"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="waybel33.html#E1:32"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:32"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel28.html#K4">xi</a> <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:32"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:32"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="waybel12.html#D1">WAYBEL12:def 1</a></i>;<br/>


</div>
