<?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="waybel_0.html#NM8">LATTICE</a>;<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/><b>let </b><font color="Maroon">p</font> be  <a href="waybel28.html#V1">greater_or_equal_to_id</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>;<br/>





<b>set </b><font color="Maroon">M</font> = <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font>;<br/>
<a NAME="E1:29"/><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</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><a class="ref" href="waybel28.html#T7">WAYBEL28:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">rM</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"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font>;<br/>
<a NAME="E4:29"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</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>
 
;<br/>
<a NAME="E5:29"/><b>then </b>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</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>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>
<a NAME="E6:29"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">C</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>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:29"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">C</font> <a href="subset_1.html#K6">\</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"><span class="p3">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font> &amp; <font color="Maroon">C</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"><span class="p3">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <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/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">C</font> <a href="subset_1.html#K6">\</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"><span class="p3">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span>, <font color="Maroon">A</font> = <font color="Maroon">C</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"><span class="p3">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> 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/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E10:29"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">F</font> is <a href="waybel_7.html#V2">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T26">WAYBEL_7:26</a></i>;<br/>
<i><font color="Green">E54</font></i>: <font color="Maroon">D</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">A</font> = 
<font color="Maroon">D</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T17">YELLOW_1:17</a></i>
<br/>.= 
<font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><a NAME="E3:29_2"/>
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="E4:29_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/><a NAME="E5:29_2"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/><a NAME="E8:29_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</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/>;<br/><b>then reconsider </b><font color="Maroon">dD</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">D</font></span><a href="domain_1.html#K1">]</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> <b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon">dD'</font> = <font color="Maroon">dD</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <b>by </b><i><a class="ref" href="waybel28.html#T7">WAYBEL28:7</a></i>;<br/><a NAME="E11:29_2"/>
 ex <font color="Olive">i</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> st <br/>for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>  st <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">i</font> holds <br/><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">dD</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span><b> such that </b><br/><a NAME="E2:29_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dD'</font>
 ;<br/>
<a NAME="E3:29_2_1"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>  st <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">dD</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:29_2_1_1"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j</font> 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> <b>by </b><i><a class="ref" href="waybel28.html#T7">WAYBEL28:7</a></i>;<br/>
<a NAME="E3:29_2_1_1"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">j'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="waybel28.html#D1">WAYBEL28:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">j'</font> = <font color="Maroon">j'</font> 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/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> 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> <b>by </b><i><a class="ref" href="waybel28.html#T7">WAYBEL28:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i'</font> 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="E7:29_2_1_1"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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>,the <a href="orders_2.html#U1">InternalRel</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><a class="ref" href="waybel28.html#D2">WAYBEL28:def 2</a></i>;<br/>
<a NAME="E8:29_2_1_1"/><b>then </b>
<font color="Maroon">i'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">j'</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<b>hence </b><a NAME="E9:29_2_1_1"/>
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">dD</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">i</font>
;<br/>

<b>thus </b><a NAME="E4:29_2_1"/>
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>  st <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">dD</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><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"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span><b> such that </b><br/><a NAME="E13:29_2"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>  st <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font> holds <br/><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Olive">j</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">dD</font>
 ;<br/><a NAME="E14:29_2"/>
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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>,the <a href="orders_2.html#U1">InternalRel</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><a class="ref" href="waybel28.html#D2">WAYBEL28:def 2</a></i>;<br/><a NAME="E15:29_2"/><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="waybel28.html#K2">*</a> <font color="Maroon">p</font> is <a href="orders_2.html#V2">reflexive</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T12">WAYBEL_8:12</a></i>;<br/><a NAME="E16:29_2"/><b>then </b>
<font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D1">YELLOW_0:def 1</a></i>;<br/><a NAME="E17:29_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">dD</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> 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="E19:29_2"/>
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <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/><a NAME="E20:29_2"/><b>then </b>
<font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</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/><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="E22:29_2"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <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> )
 ;<br/><a NAME="E23:29_2"/><i><font color="Green">E131</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E24:29_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">dD</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E25:29_2"/><b>then </b><i><font color="Green">E132</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i>, , , <a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/><a NAME="E26:29_2"/><i><font color="Green">E133</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</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><a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default">the <a href="waybel_0.html#U1">mapping</a> of <span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="waybel28.html#D2">WAYBEL28:def 2</a></i>
<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="Maroon">G</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>
;<br/><a NAME="E28:29_2"/><b>then </b><i><font color="Green">E134</font></i>: 
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="waybel_0.html#K2">.</a> <font color="Maroon">i</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"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E29:29_2"/>
the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="yellow_6.html#K3">.</a> <font color="Maroon">i</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"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E30:29_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E13:29"/><b>then </b><i><font color="Green">E135</font></i>: 
( <font color="Maroon">A</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 <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> &amp; <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#D2">WAYBEL_7:def 2</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
 <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</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"><span class="p3">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</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/>
<a NAME="E15:29"/><b>then </b><i><font color="Green">E136</font></i>: 
 <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A'</font>
 
<b>by </b><i>, <a class="ref" href="waybel_7.html#T3">WAYBEL_7:3</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<a NAME="E16:29"/><i><font color="Green">E155</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A'</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">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E17:29"/><i><font color="Green">E156</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">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </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="E18:29"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A'</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_4.html#T1">YELLOW_4:1</a></i>;<br/>
<b>hence </b><a NAME="E19:29"/>
 <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel28.html#K2">*</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D2">YELLOW_0:def 2</a></i>;<br/>


</div>
