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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">V</font> be  <a href="waybel_0.html#V13">upper</a> <a href="waybel_6.html#V1">Open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V13">upper</a> <a href="waybel_6.html#V1">Open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font> <b>by </b><i/>;<br/>
<b>given </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="waybel12.html#M1" target="_self">GeneratorSet</a> of <font color="Maroon">F</font><b> such that </b><a NAME="E5:54"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A</font> is <a href="card_4.html#V1">countable</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> </span>}</span>  as   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font> <b>by </b><i>, , </i>;<br/>
<a NAME="E7:54"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:54_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:54_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">t</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42">YELLOW_4:42</a></i>;<br/>
<a NAME="E4:54_1"/>
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:54_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">s</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">t</font>
 <b>and </b><br/><a NAME="E4:54_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<a NAME="E5:54_1_1"/>
<font color="Maroon">t</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="waybel12.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E6:54_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:54_1"><i><font color="Green">E32</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">A</font><b> such that </b><br/><a NAME="E9:54"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ,<font color="Maroon">L</font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:54"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E13:54"/><i><font color="Green">E94</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">AA</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E15:54"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon">gn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font></span>)</span> where <font color="Olive">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> </span>}</span> , <font color="Maroon">gn1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<a NAME="E3:54_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">gn</font>,<font color="Maroon">L</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">gn1</font>,<font color="Maroon">L</font> )
 
<b>by </b><i>, <a class="ref" href="waybel12.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:54_2"/><i><font color="Green">E110</font></i>: 
(  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">gn</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">gn1</font>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T55">YELLOW_0:55</a></i>;<br/>
<a NAME="E5:54_2"/>
<font color="Maroon">gn</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">gn1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</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="E1:54_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">gn</font>
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:54_2_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>
<a NAME="E5:54_2_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<b>hence </b><a NAME="E6:54_2_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">gn1</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:54_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T3" target="_self">Th3</a>, <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:54"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">AA</font> is    <a href="waybel12.html#M1" target="_self">GeneratorSet</a> of <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#D2" target="_self">Def2</a>, , </i>;<br/>
<a NAME="E17:54"/><b>then </b><i><font color="Green">E115</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">AA</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:54"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">x1</font>, <font color="Olive">y1</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font> ex <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Olive">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Olive">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Olive">z1</font> );<br/>
<a NAME="E18:54"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>;<br/>



<a NAME="E1:54_3"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:54_3"/>
<font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T2" target="_self">Th2</a>, <a class="txt" href="waybel12.html#E2:54"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E3:54_3"/><b>then </b>
<font color="Maroon">AA</font> <a href="tarski.html#R1">c=</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="E4:54_3"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T4" target="_self">Th4</a></i>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:54_3"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font>
 <b>and </b><br/><a NAME="E7:54_3"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g1</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 ;<br/>
<a NAME="E8:54_3"/>
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g1</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:54_3"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> <a href="yellow_4.html#K4">"/\"</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">g1</font></span><a href="domain_1.html#K6">}</a></span>
 
;<br/>
<a NAME="E10:54_3"/><b>then </b>
 ex <font color="Olive">y1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">g1</font> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="waybel_6.html#D1">WAYBEL_6:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:54_3"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">V1</font><b> such that </b><br/><a NAME="E20:54"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 <b>and </b><br/><a NAME="E21:54"/><i><font color="Green">E124</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="ref" href="waybel12.html#T3" target="_self">Th3</a>);<br/></i>
<a NAME="E22:54"/><i><font color="Green">E125</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E23:54"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon">O</font> =  <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">R</font></span>)</span>;<br/>
<a NAME="E25:54"/><i><font color="Green">E128</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:54_4"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:54_4"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<a NAME="E4:54_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T8" target="_self">Th8</a>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<a NAME="E5:54_4"/>
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E6:54_4"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T8" target="_self">Th8</a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E26:54"/><i><font color="Green">E132</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <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="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:54_5"/><i><font color="Green">E133</font></i>: 
( <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">a</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Olive">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">t</font>;<br/>
<a NAME="E2:54_5"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<a NAME="E3:54_5"/><i><font color="Green">E135</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:54_5_1"/><i><font color="Green">E136</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">s</font>, <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><font color="Olive">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">t</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E2:54_5_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<a NAME="E3:54_5_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E5:54_5_1"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_5_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_1"><b>suppose </b></a><a NAME="E1:54_5_1_1_1"/>
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:54_5_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:54_5_1_1_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T12" target="_self">Th12</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_5_1_1_2"><b>suppose </b></a><a NAME="E1:54_5_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_5_1_1_2"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:54_5"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:54_5"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:54"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; not <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:54_6"/><i><font color="Green">E142</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_6"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E4:54_6"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">m</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:54_6"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E8:54_6"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_6_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1_1"><b>suppose </b></a><a NAME="E1:54_6_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_6_1_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="waybel12.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_6_1_2"><b>suppose </b></a><a NAME="E1:54_6_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_6_1_2"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="waybel12.html#T12" target="_self">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E28:54"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">R</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">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T62">WAYBEL_0:62</a></i>;<br/>
<a NAME="E29:54"/>
 <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">R</font></span>)</span> is <a href="waybel_6.html#V1">Open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="waybel_6.html#D1">WAYBEL_6:def 1</a><br/>

<b>assume </b><a NAME="E1:54_7"/>
<font color="Maroon">x</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">R</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:54_7"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:54_7"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>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">R</font><b> such that </b><br/><a NAME="E6:54_7"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E7:54_7"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_7_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_7_1_1"><b>suppose </b></a><a NAME="E1:54_7_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:54_7_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T8" target="_self">Th8</a>, , </i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54_7_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E5:54_7_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="waybel12.html#T12" target="_self">Th12</a></i>;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:54_7_1_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E9:54_7_1_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E10:54_7_1_1"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E11:54_7_1_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">y1</font>
;<br/><a NAME="E12:54_7_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="waybel12.html#T16" target="_self">Th16</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E13:54_7_1_1"/>
<font color="Maroon">y1</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">R</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:54_7_1_1"/>
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E15:54_7_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T18" target="_self">Th18</a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><b>hence </b><a NAME="E16:54_7_1_1"/>
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T13" target="_self">Th13</a>, <a class="ref" href="waybel12.html#T14" target="_self">Th14</a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_7_1_2"><b>suppose </b></a><a NAME="E1:54_7_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:54_7_1_2"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D12">YELLOW_0:def 12</a></i>;<br/><a NAME="E3:54_7_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><a NAME="E4:54_7_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T12" target="_self">Th12</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:54_7_1_2"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel12.html#T14" target="_self">Th14</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:54_7_1_2"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E10:54_7_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T14" target="_self">Th14</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="waybel12.html#T16" target="_self">Th16</a></i>;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E13:54_7_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E14:54_7_1_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E15:54_7_1_2"/>
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E16:54_7_1_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">y1</font>
;<br/><a NAME="E17:54_7_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="waybel12.html#T18" target="_self">Th18</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E18:54_7_1_2"/>
<font color="Maroon">y1</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">R</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E19:54_7_1_2"/>
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/><a NAME="E20:54_7_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><a NAME="E21:54_7_1_2"/>
<font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T13" target="_self">Th13</a>, <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><b>hence </b><a NAME="E22:54_7_1_2"/>
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T20" target="_self">Th20</a>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</font> =  <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">R</font></span>)</span> as  <a href="waybel_6.html#V1">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">O</font>
;<br/>

<b>thus </b><a NAME="E31:54"/>
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:54_8"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">q</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">O</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:54_8"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font>
 <b>and </b><br/><a NAME="E5:54_8"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K12">fininfs</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16">WAYBEL_0:def 16</a></i>;<br/>
<b>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">R</font><b> such that </b><br/><a NAME="E7:54_8"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E8:54_8"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Y</font>,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_8_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_8_1_1"><b>suppose </b></a><a NAME="E1:54_8_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:54_8_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T8" target="_self">Th8</a>, , </i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54_8_1_1"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E5:54_8_1_1"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="waybel12.html#T12" target="_self">Th12</a></i>;<br/><b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:54_8_1_1"/><i><font color="Green">E168</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E9:54_8_1_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a></i>;<br/><b>thus </b><a NAME="E10:54_8_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T13" target="_self">Th13</a>, <a class="ref" href="waybel12.html#T14" target="_self">Th14</a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_8_1_2"><b>suppose </b></a><a NAME="E1:54_8_1_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:54_8_1_2"/><b>then </b><i><font color="Green">E169</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D12">YELLOW_0:def 12</a></i>;<br/><a NAME="E3:54_8_1_2"/>
<font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><a NAME="E4:54_8_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T12" target="_self">Th12</a>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/><b>hence </b><a NAME="E5:54_8_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E32:54"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T4" target="_self">Th4</a>, , </i>;<br/>

<a NAME="E33:54"/><i><font color="Green">E170</font></i>: 
<font color="Maroon">AA</font> <a href="yellow_4.html#R4">is_coarser_than</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="yellow_4.html#D2">YELLOW_4:def 2</a><br/>

<b>assume </b><a NAME="E1:54_9"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_9"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:54_9"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">y1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">V1</font>, <font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:54_9"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:54_9"/><i><font color="Green">E173</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E9:54_9"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E10:54_9"/><i><font color="Green">E175</font></i>: 
<font color="Maroon">y1</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E8"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<a NAME="E11:54_9"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E9"><i><font color="Green">Lemma45</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:54_9"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E13:54_9"/>
<font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T14" target="_self">Th14</a>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<a NAME="E14:54_9"/><b>then </b><i><font color="Green">E178</font></i>: 
<font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#T18" target="_self">Th18</a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<a NAME="E15:54_9"/>
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="waybel12.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E16:54_9"/><b>then </b>
<font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E17:54_9"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="waybel12.html#T12" target="_self">Th12</a>, <a class="ref" href="waybel12.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:54"/>
<font color="Maroon">R</font> <a href="yellow_4.html#R4">is_coarser_than</a> <font color="Maroon">O</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E35:54"/><b>then </b>
<font color="Maroon">AA</font> <a href="yellow_4.html#R4">is_coarser_than</a> <font color="Maroon">O</font>
 
<b>by </b><i>, <a class="ref" href="yellow_4.html#T7">YELLOW_4:7</a></i>;<br/>
<a NAME="E36:54"/><b>then </b>
<font color="Maroon">AA</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T8">YELLOW_4:8</a></i>;<br/>
<b>hence </b><a NAME="E37:54"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T62">WAYBEL_0:62</a></i>;<br/>


</div>
