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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c2">V</font> be  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">V1</font> = <font color="Maroon" title="c2">V</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel12.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> <b>by </b><i><a class="txt" href="waybel12.html#E2:52"><i><font color="Green" title="E31">A2</font></i></a></i>;<br/>
<b>given </b><font color="Maroon" title="c7">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font><b> such that </b><a NAME="E6:52"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c7">A</font> is <a href="card_4.html#V1" title="CARD_4:attr.1">countable</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">G</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> </span>}</span>  as   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="waybel12.html#T27" target="_self" title="WAYBEL12:th.27">Th27</a>, <a class="ref" href="waybel12.html#T28" target="_self" title="WAYBEL12:th.28">Th28</a>, <a class="ref" href="waybel12.html#T29" target="_self" title="WAYBEL12:th.29">Th29</a></i>;<br/>
<a NAME="E8:52"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green" title="E34">A5</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c9">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E3:52_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">s</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#T42" title="YELLOW_4:th.42">YELLOW_4:42</a></i>;<br/>
<a NAME="E4:52_1"/>
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_1_1"/>
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c10">s</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:52_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">t</font>
 <b>and </b><br/><a NAME="E4:52_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:52_1"><i><font color="Green" title="E35">A6</font></i></a></i>;<br/>
<a NAME="E5:52_1_1"/>
<font color="Maroon" title="c12">t</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:52_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="waybel12.html#E4:52_1_1"><i><font color="Green" title="E37">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:52_1_1"/>
<font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:52"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="waybel12.html#E3:52_1_1"><i><font color="Green" title="E36">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:52_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c7">A</font><b> such that </b><br/><a NAME="E10:52"/><i><font color="Green" title="E34">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E6:52"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="waybel12.html#T4" target="_self" title="WAYBEL12:th.4">Th4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> =  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 </span>}</span> ,<font color="Maroon" title="c1">L</font>;<br/>
<b>consider </b><font color="Maroon" title="c11">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E13:52"/><i><font color="Green" title="E35">A10</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E14:52"/><i><font color="Green" title="E36">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">AA</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">g</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></i>;<br/>
<a NAME="E16:52"/><i><font color="Green" title="E37">A12</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds <br/><font color="Olive" title="b3">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:52_2"/><i><font color="Green" title="E38">A13</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c16">gn</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">m</font></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> </span>}</span> , <font color="Maroon" title="c17">gn1</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel12.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E3:52_2"/><i><font color="Green" title="E39">A14</font></i>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c16">gn</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c17">gn1</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E13:52"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="waybel12.html#E1:52_2"><i><font color="Green" title="E38">A13</font></i></a></i>;<br/>
<a NAME="E4:52_2"/><i><font color="Green" title="E40">A15</font></i>: 
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c16">gn</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c17">gn1</font>,<font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T55" title="YELLOW_0:th.55">YELLOW_0:55</a></i>;<br/>
<a NAME="E5:52_2"/>
<font color="Maroon" title="c16">gn</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">gn1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_2_1"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">gn</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:52_2_1"/><i><font color="Green" title="E41">A16</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">k</font>
 <b>and </b><br/><a NAME="E4:52_2_1"/><i><font color="Green" title="E42">A17</font></i>: 
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 ;<br/>
<a NAME="E5:52_2_1"/>
<font color="Maroon" title="c19">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="waybel12.html#E4:52_2_1"><i><font color="Green" title="E42">A17</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<b>hence </b><a NAME="E6:52_2_1"/>
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">gn1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:52_2_1"><i><font color="Green" title="E41">A16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:52_2"/>
<font color="Maroon" title="c15">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:52_2"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="waybel12.html#E4:52_2"><i><font color="Green" title="E40">A15</font></i></a>, <a class="ref" href="yellow_0.html#T35" title="YELLOW_0:th.35">YELLOW_0:35</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:52"/><i><font color="Green" title="E38">A18</font></i>: 
<font color="Maroon" title="c12">AA</font> is    <a href="waybel12.html#M1" title="WAYBEL12:mode.1">GeneratorSet</a> of <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E10:52"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="waybel12.html#E13:52"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="waybel12.html#T36" target="_self" title="WAYBEL12:th.36">Th36</a></i>;<br/>
<a NAME="E18:52"/><b>then </b><i><font color="Green" title="E39">A19</font></i>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c12">AA</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="ref" href="waybel12.html#D3" target="_self" title="WAYBEL12:def.3">Def3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">y1</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> ex <font color="Olive" title="b3">z1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">$1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Olive" title="b1">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Olive" title="b3">z1</font> );<br/>
<a NAME="E19:52"/><i><font color="Green" title="E40">A20</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>  ex <font color="Olive" title="b3">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>;<br/>



<a NAME="E1:52_3"/><i><font color="Green" title="E41">A21</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E14:52"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E2:52_3"/>
<font color="Maroon" title="c12">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E17:52"><i><font color="Green" title="E38">A18</font></i></a>, <a class="txt" href="waybel12.html#E39"><i><font color="Green" title="E24">Lm4</font></i></a></i>;<br/>
<a NAME="E3:52_3"/><b>then </b>
<font color="Maroon" title="c12">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E8:52"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:52_3"/><b>then </b>
<font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">G</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E1:52_3"><i><font color="Green" title="E41">A21</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">g1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E6:52_3"/><i><font color="Green" title="E42">A22</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g1</font>
 <b>and </b><br/><a NAME="E7:52_3"/><i><font color="Green" title="E43">A23</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 ;<br/>
<a NAME="E8:52_3"/>
<font color="Maroon" title="c15">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:52_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> <a href="yellow_4.html#K4" title="YELLOW_4:func.4">"/\"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c15">g1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
;<br/>
<a NAME="E10:52_3"/><b>then </b>
 ex <font color="Olive" title="b1">y1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c14">x</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c15">g1</font> )
 
<b>by </b><i><a class="txt" href="waybel12.html#E7:52_3"><i><font color="Green" title="E43">A23</font></i></a>, <a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:52_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c14">x</font>,<font color="Olive" title="b1">y</font>]
 <b>by </b><i><a class="txt" href="waybel12.html#E6:52_3"><i><font color="Green" title="E42">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c13">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c5">V1</font><b> such that </b><br/><a NAME="E21:52"/><i><font color="Green" title="E41">A24</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font>
 <b>and </b><br/><a NAME="E22:52"/><i><font color="Green" title="E42">A25</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2" title="RECDEF_1:sch.2">RECDEF_1:sch 2</a>(<a class="txt" href="waybel12.html#E19:52"><i><font color="Green" title="E40">A20</font></i></a>);<br/></i>
<a NAME="E23:52"/><i><font color="Green" title="E43">A26</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E24:52"/><b>then </b><i><font color="Green" title="E44">A27</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">h</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">R</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c13">h</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">O</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>;<br/>
<a NAME="E26:52"/><i><font color="Green" title="E45">A28</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:52_4"/><i><font color="Green" title="E46">A29</font></i>: 
( <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c21">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:52_4"/><i><font color="Green" title="E47">A30</font></i>: 
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c20">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E22:52"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/>
<a NAME="E4:52_4"/><i><font color="Green" title="E48">A31</font></i>: 
<font color="Maroon" title="c20">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E3:52_4"><i><font color="Green" title="E47">A30</font></i></a>, <a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a></i>;<br/>
<a NAME="E5:52_4"/>
<font color="Maroon" title="c19">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c21">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c19">x1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<b>hence </b><a NAME="E6:52_4"/>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:52_4"><i><font color="Green" title="E46">A29</font></i></a>, <a class="txt" href="waybel12.html#E3:52_4"><i><font color="Green" title="E47">A30</font></i></a>, <a class="txt" href="waybel12.html#E4:52_4"><i><font color="Green" title="E48">A31</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:52"/><i><font color="Green" title="E46">A32</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">m</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c18">n</font>, <font color="Maroon" title="c19">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:52_5"/><i><font color="Green" title="E47">A33</font></i>: 
( <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">x</font> &amp; <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 &amp; <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><font color="Olive" title="b2">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">t</font>;<br/>
<a NAME="E2:52_5"/><i><font color="Green" title="E48">A34</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/>
<a NAME="E3:52_5"/><i><font color="Green" title="E49">A35</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:52_5_1"/><i><font color="Green" title="E50">A36</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">k</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">k</font> holds <br/><font color="Olive" title="b2">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">t</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c21">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c22">c</font>, <font color="Maroon" title="c23">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E2:52_5_1"/><i><font color="Green" title="E51">A37</font></i>: 
( <font color="Maroon" title="c23">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">a</font> &amp; <font color="Maroon" title="c22">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c21">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/>

<a NAME="E3:52_5_1"/>
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E23:52"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">s</font> = <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E5:52_5_1"/><i><font color="Green" title="E52">A38</font></i>: 
<font color="Maroon" title="c22">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c24">s</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E26:52"><i><font color="Green" title="E45">A28</font></i></a>, <a class="txt" href="waybel12.html#E2:52_5_1"><i><font color="Green" title="E51">A37</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_5_1_1"/>
( <font color="Maroon" title="c21">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">k</font> or <font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="waybel12.html#E2:52_5_1"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_1_1_1"><b>suppose </b></a><a NAME="E1:52_5_1_1_1"/>
<font color="Maroon" title="c21">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">k</font>
 ;<br/><div class="add"><a NAME="E2:52_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c24">s</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c23">d</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:52_5_1"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="waybel12.html#E2:52_5_1"><i><font color="Green" title="E51">A37</font></i></a></i>;<br/><b>hence </b><a NAME="E3:52_5_1_1_1"/>
<font color="Maroon" title="c22">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c23">d</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E5:52_5_1"><i><font color="Green" title="E52">A38</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_5_1_1_2"><b>suppose </b></a><a NAME="E1:52_5_1_1_2"/>
<font color="Maroon" title="c21">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_5_1_1_2"/>
<font color="Maroon" title="c22">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c23">d</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E2:52_5_1"><i><font color="Green" title="E51">A37</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:52_5"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="waybel12.html#E2:52_5"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="waybel12.html#E3:52_5"><i><font color="Green" title="E49">A35</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:52_5"/>
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:52_5"><i><font color="Green" title="E47">A33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:52"/><i><font color="Green" title="E47">A39</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; not <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font>, <font color="Maroon" title="c17">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:52_6"/><i><font color="Green" title="E48">A40</font></i>: 
( <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_6"/><i><font color="Green" title="E49">A41</font></i>: 
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h</font>
 <b>and </b><br/><a NAME="E4:52_6"/><i><font color="Green" title="E50">A42</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">n</font> = <font color="Maroon" title="c18">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="waybel12.html#E3:52_6"><i><font color="Green" title="E49">A41</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">m</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:52_6"/><i><font color="Green" title="E51">A43</font></i>: 
<font color="Maroon" title="c20">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h</font>
 <b>and </b><br/><a NAME="E8:52_6"/><i><font color="Green" title="E52">A44</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E1:52_6"><i><font color="Green" title="E48">A40</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">m</font> = <font color="Maroon" title="c20">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="waybel12.html#E7:52_6"><i><font color="Green" title="E51">A43</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_6_1"/>
( <font color="Maroon" title="c21">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font> or <font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_6_1_1"><b>suppose </b></a><a NAME="E1:52_6_1_1"/>
<font color="Maroon" title="c21">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_6_1_1"/>
( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E27:52"><i><font color="Green" title="E46">A32</font></i></a>, <a class="txt" href="waybel12.html#E4:52_6"><i><font color="Green" title="E50">A42</font></i></a>, <a class="txt" href="waybel12.html#E8:52_6"><i><font color="Green" title="E52">A44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_6_1_2"><b>suppose </b></a><a NAME="E1:52_6_1_2"/>
<font color="Maroon" title="c19">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:52_6_1_2"/>
( <font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">y</font> or <font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E27:52"><i><font color="Green" title="E46">A32</font></i></a>, <a class="txt" href="waybel12.html#E4:52_6"><i><font color="Green" title="E50">A42</font></i></a>, <a class="txt" href="waybel12.html#E8:52_6"><i><font color="Green" title="E52">A44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E29:52"/><i><font color="Green" title="E48">A45</font></i>: 
<font color="Maroon" title="c14">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T62" title="WAYBEL_0:th.62">WAYBEL_0:62</a></i>;<br/>
<a NAME="E30:52"/>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a><br/>

<b>assume </b><a NAME="E1:52_7"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c17">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:52_7"/><i><font color="Green" title="E49">A46</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <b>and </b><br/><a NAME="E4:52_7"/><i><font color="Green" title="E50">A47</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">Y</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c14">R</font><b> such that </b><br/><a NAME="E6:52_7"/><i><font color="Green" title="E51">A48</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c18">Y</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E7:52_7"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c18">Y</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:52_7"><i><font color="Green" title="E50">A47</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_7_1"/>
( <font color="Maroon" title="c18">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c18">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_7_1_1"><b>suppose </b></a><a NAME="E1:52_7_1_1"/>
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_7_1_1"/><b>then </b>
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">Y</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E28:52"><i><font color="Green" title="E47">A39</font></i></a>, <a class="txt" href="waybel12.html#E6:52_7"><i><font color="Green" title="E51">A48</font></i></a>, <a class="ref" href="waybel12.html#T31" target="_self" title="WAYBEL12:th.31">Th31</a></i>;<br/><b>then consider </b><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:52_7_1_1"/><i><font color="Green" title="E52">A49</font></i>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h</font>
 <b>and </b><br/><a NAME="E5:52_7_1_1"/><i><font color="Green" title="E53">A50</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">n</font> = <font color="Maroon" title="c19">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="waybel12.html#E4:52_7_1_1"><i><font color="Green" title="E52">A49</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">x1</font>, <font color="Maroon" title="c22">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c23">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:52_7_1_1"/><i><font color="Green" title="E54">A51</font></i>: 
<font color="Maroon" title="c21">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">n</font>
 <b>and </b><br/><a NAME="E9:52_7_1_1"/><i><font color="Green" title="E55">A52</font></i>: 
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E10:52_7_1_1"/>
<font color="Maroon" title="c23">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E11:52_7_1_1"/><i><font color="Green" title="E56">A53</font></i>: 
<font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c21">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c23">z1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E22:52"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c22">y1</font>
;<br/><a NAME="E12:52_7_1_1"/>
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E23:52"><i><font color="Green" title="E43">A26</font></i></a>, <a class="txt" href="waybel12.html#E9:52_7_1_1"><i><font color="Green" title="E55">A52</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E13:52_7_1_1"/>
<font color="Maroon" title="c22">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel12.html#E29:52"><i><font color="Green" title="E48">A45</font></i></a></i>;<br/><a NAME="E14:52_7_1_1"/>
<font color="Maroon" title="c21">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c23">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c21">x1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E15:52_7_1_1"/><b>then </b>
<font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c21">x1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E11:52_7_1_1"><i><font color="Green" title="E56">A53</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><b>hence </b><a NAME="E16:52_7_1_1"/>
<font color="Maroon" title="c22">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:52_7"><i><font color="Green" title="E49">A46</font></i></a>, <a class="txt" href="waybel12.html#E5:52_7_1_1"><i><font color="Green" title="E53">A50</font></i></a>, <a class="txt" href="waybel12.html#E8:52_7_1_1"><i><font color="Green" title="E54">A51</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_7_1_2"><b>suppose </b></a><a NAME="E1:52_7_1_2"/>
<font color="Maroon" title="c18">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_7_1_2"/><b>then </b><i><font color="Green" title="E52">A54</font></i>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E6:52_7"><i><font color="Green" title="E51">A48</font></i></a>, <a class="ref" href="yellow_0.html#D12" title="YELLOW_0:def.12">YELLOW_0:def 12</a></i>;<br/><a NAME="E3:52_7_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></i>;<br/><a NAME="E4:52_7_1_2"/><b>then </b><i><font color="Green" title="E53">A55</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E3:52_7"><i><font color="Green" title="E49">A46</font></i></a>, <a class="txt" href="waybel12.html#E2:52_7_1_2"><i><font color="Green" title="E52">A54</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/><b>consider </b><font color="Maroon" title="c19">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:52_7_1_2"/><i><font color="Green" title="E54">A56</font></i>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c20">b</font> = <font color="Maroon" title="c19">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="txt" href="waybel12.html#E6:52_7_1_2"><i><font color="Green" title="E54">A56</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:52_7_1_2"/><i><font color="Green" title="E55">A57</font></i>: 
<font color="Maroon" title="c21">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h</font>
 <b>and </b><br/><a NAME="E10:52_7_1_2"/>
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E6:52_7_1_2"><i><font color="Green" title="E54">A56</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">n</font> = <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="waybel12.html#E9:52_7_1_2"><i><font color="Green" title="E55">A57</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c23">x1</font>, <font color="Maroon" title="c24">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c25">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E13:52_7_1_2"/>
<font color="Maroon" title="c23">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font>
 <b>and </b><br/><a NAME="E14:52_7_1_2"/><i><font color="Green" title="E56">A58</font></i>: 
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E15:52_7_1_2"/>
<font color="Maroon" title="c25">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E16:52_7_1_2"/><i><font color="Green" title="E57">A59</font></i>: 
<font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c23">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c25">z1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E22:52"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/><b>take </b>
<font color="Maroon" title="c24">y1</font>
;<br/><a NAME="E17:52_7_1_2"/>
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E23:52"><i><font color="Green" title="E43">A26</font></i></a>, <a class="txt" href="waybel12.html#E14:52_7_1_2"><i><font color="Green" title="E56">A58</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E18:52_7_1_2"/>
<font color="Maroon" title="c24">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel12.html#E29:52"><i><font color="Green" title="E48">A45</font></i></a></i>;<br/><a NAME="E19:52_7_1_2"/>
<font color="Maroon" title="c23">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c25">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c23">x1</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/><a NAME="E20:52_7_1_2"/><b>then </b><i><font color="Green" title="E58">A60</font></i>: 
<font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c23">x1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E16:52_7_1_2"><i><font color="Green" title="E57">A59</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">WAYBEL_3:2</a></i>;<br/><a NAME="E21:52_7_1_2"/>
<font color="Maroon" title="c23">x1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:52_7_1_2"><i><font color="Green" title="E53">A55</font></i></a>, <a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></i>;<br/><b>hence </b><a NAME="E22:52_7_1_2"/>
<font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c16">x</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E20:52_7_1_2"><i><font color="Green" title="E58">A60</font></i></a>, <a class="ref" href="waybel_3.html#T2" title="WAYBEL_3:th.2">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" title="c16">O</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font></span>)</span> as  <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c16">O</font>
;<br/>

<b>thus </b><a NAME="E32:52"/>
<font color="Maroon" title="c16">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:52_8"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c18">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">O</font> ;<br/>
<b>consider </b><font color="Maroon" title="c19">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:52_8"/><i><font color="Green" title="E49">A61</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c18">q</font>
 <b>and </b><br/><a NAME="E5:52_8"/><i><font color="Green" title="E50">A62</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c14">R</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">Y</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c14">R</font><b> such that </b><br/><a NAME="E7:52_8"/><i><font color="Green" title="E51">A63</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c20">Y</font>,<font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E8:52_8"/>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c20">Y</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E5:52_8"><i><font color="Green" title="E50">A62</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_8_1"/>
( <font color="Maroon" title="c20">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_8_1_1"><b>suppose </b></a><a NAME="E1:52_8_1_1"/>
<font color="Maroon" title="c20">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_8_1_1"/><b>then </b>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Y</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E28:52"><i><font color="Green" title="E47">A39</font></i></a>, <a class="txt" href="waybel12.html#E7:52_8"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="waybel12.html#T31" target="_self" title="WAYBEL12:th.31">Th31</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:52_8_1_1"/><i><font color="Green" title="E52">A64</font></i>: 
<font color="Maroon" title="c21">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h</font>
 <b>and </b><br/><a NAME="E5:52_8_1_1"/><i><font color="Green" title="E53">A65</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">n</font> = <font color="Maroon" title="c21">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="waybel12.html#E4:52_8_1_1"><i><font color="Green" title="E52">A64</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c23">x1</font>, <font color="Maroon" title="c24">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c25">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:52_8_1_1"/><i><font color="Green" title="E54">A66</font></i>: 
<font color="Maroon" title="c23">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">n</font>
 <b>and </b><br/><a NAME="E9:52_8_1_1"/>
( <font color="Maroon" title="c24">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c25">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c24">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c23">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c25">z1</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E22:52"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/><b>thus </b><a NAME="E10:52_8_1_1"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:52_8"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="waybel12.html#E5:52_8_1_1"><i><font color="Green" title="E53">A65</font></i></a>, <a class="txt" href="waybel12.html#E8:52_8_1_1"><i><font color="Green" title="E54">A66</font></i></a>, <a class="ref" href="waybel_0.html#D20" title="WAYBEL_0:def.20">WAYBEL_0:def 20</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_8_1_2"><b>suppose </b></a><a NAME="E1:52_8_1_2"/>
<font color="Maroon" title="c20">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:52_8_1_2"/><b>then </b><i><font color="Green" title="E52">A67</font></i>: 
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E7:52_8"><i><font color="Green" title="E51">A63</font></i></a>, <a class="ref" href="yellow_0.html#D12" title="YELLOW_0:def.12">YELLOW_0:def 12</a></i>;<br/><a NAME="E3:52_8_1_2"/>
<font color="Maroon" title="c18">q</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T45" title="YELLOW_0:th.45">YELLOW_0:45</a></i>;<br/><a NAME="E4:52_8_1_2"/><b>then </b>
<font color="Maroon" title="c18">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E4:52_8"><i><font color="Green" title="E49">A61</font></i></a>, <a class="txt" href="waybel12.html#E2:52_8_1_2"><i><font color="Green" title="E52">A67</font></i></a>, <a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/><b>hence </b><a NAME="E5:52_8_1_2"/>
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="waybel12.html#T12" target="_self" title="WAYBEL12:th.12">Th12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E33:52"/>
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">O</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E21:52"><i><font color="Green" title="E41">A24</font></i></a>, <a class="txt" href="waybel12.html#E24:52"><i><font color="Green" title="E44">A27</font></i></a>, <a class="txt" href="waybel12.html#E29:52"><i><font color="Green" title="E48">A45</font></i></a></i>;<br/>

<a NAME="E34:52"/><i><font color="Green" title="E49">A68</font></i>: 
<font color="Maroon" title="c12">AA</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c14">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_4.html#D2" title="YELLOW_4:def.2">YELLOW_4:def 2</a><br/>

<b>assume </b><a NAME="E1:52_9"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">AA</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c18">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_9"/><i><font color="Green" title="E50">A69</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">g</font>
 <b>and </b><br/><a NAME="E4:52_9"/><i><font color="Green" title="E51">A70</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">x</font> = <font color="Maroon" title="c18">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="waybel12.html#E3:52_9"><i><font color="Green" title="E50">A69</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c21">y1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">V1</font>, <font color="Maroon" title="c22">z1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E7:52_9"/>
<font color="Maroon" title="c20">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">x</font>
 <b>and </b><br/><a NAME="E8:52_9"/><i><font color="Green" title="E52">A71</font></i>: 
<font color="Maroon" title="c21">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E9:52_9"/><i><font color="Green" title="E53">A72</font></i>: 
<font color="Maroon" title="c22">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>and </b><br/><a NAME="E10:52_9"/><i><font color="Green" title="E54">A73</font></i>: 
<font color="Maroon" title="c21">y1</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c20">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c22">z1</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E22:52"><i><font color="Green" title="E42">A25</font></i></a></i>;<br/>
<a NAME="E11:52_9"/><i><font color="Green" title="E55">A74</font></i>: 
<font color="Maroon" title="c13">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E23:52"><i><font color="Green" title="E43">A26</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:52_9"/><i><font color="Green" title="E56">A75</font></i>: 
<font color="Maroon" title="c20">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c22">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c22">z1</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a></i>;<br/>
<a NAME="E13:52_9"/>
<font color="Maroon" title="c21">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c20">x1</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c22">z1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E10:52_9"><i><font color="Green" title="E54">A73</font></i></a>, <a class="ref" href="waybel_3.html#T1" title="WAYBEL_3:th.1">WAYBEL_3:1</a></i>;<br/>
<a NAME="E14:52_9"/><b>then </b><i><font color="Green" title="E57">A76</font></i>: 
<font color="Maroon" title="c21">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c22">z1</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E12:52_9"><i><font color="Green" title="E56">A75</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<a NAME="E15:52_9"/>
<font color="Maroon" title="c22">z1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E16:52"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="waybel12.html#E4:52_9"><i><font color="Green" title="E51">A70</font></i></a>, <a class="txt" href="waybel12.html#E9:52_9"><i><font color="Green" title="E53">A72</font></i></a></i>;<br/>
<a NAME="E16:52_9"/><b>then </b>
<font color="Maroon" title="c21">y1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E14:52_9"><i><font color="Green" title="E57">A76</font></i></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></i>;<br/>
<b>hence </b><a NAME="E17:52_9"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">R</font> &amp; <font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c17">a</font> )
 <b>by </b><i><a class="txt" href="waybel12.html#E8:52_9"><i><font color="Green" title="E52">A71</font></i></a>, <a class="txt" href="waybel12.html#E11:52_9"><i><font color="Green" title="E55">A74</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:52"/>
<font color="Maroon" title="c14">R</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c16">O</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E29:52"><i><font color="Green" title="E48">A45</font></i></a>, <a class="ref" href="waybel12.html#T20" target="_self" title="WAYBEL12:th.20">Th20</a></i>;<br/>
<a NAME="E36:52"/><b>then </b>
<font color="Maroon" title="c12">AA</font> <a href="yellow_4.html#R4" title="YELLOW_4:pred.4">is_coarser_than</a> <font color="Maroon" title="c16">O</font>
 
<b>by </b><i><a class="txt" href="waybel12.html#E34:52"><i><font color="Green" title="E49">A68</font></i></a>, <a class="ref" href="yellow_4.html#T7" title="YELLOW_4:th.7">YELLOW_4:7</a></i>;<br/>
<a NAME="E37:52"/><b>then </b>
<font color="Maroon" title="c12">AA</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font>
 
<b>by </b><i><a class="ref" href="yellow_4.html#T8" title="YELLOW_4:th.8">YELLOW_4:8</a></i>;<br/>
<b>hence </b><a NAME="E38:52"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">O</font>
 <b>by </b><i><a class="txt" href="waybel12.html#E18:52"><i><font color="Green" title="E39">A19</font></i></a>, <a class="ref" href="waybel_0.html#T62" title="WAYBEL_0:th.62">WAYBEL_0:62</a></i>;<br/>


</div>
