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

<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W</font> =  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E4:17"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/> ex <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>,<font color="Olive">z1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:17_1"/><b>then </b>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:17_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="waybel_4.html#T53">WAYBEL_4:53</a></i>;<br/>
<a NAME="E5:17_1"/>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/>
<b>hence </b><a NAME="E6:17_1"/>
 ex <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">z</font>,<font color="Olive">z1</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">W</font>,<font color="Maroon">W</font><b> such that </b><br/><a NAME="E6:17"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> holds <br/> ex <font color="Olive">z1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Olive">z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">z</font>,<font color="Olive">z1</font>] )
 <b>from </b><i><a class="ref" href="waybel_6.html#S1" target="_self">WAYBEL_6:sch 1</a>();<br/></i>
<a NAME="E7:17"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:17"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">F</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_2"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:17_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:17_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E7:17_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z1</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><a NAME="E9:17_2"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">z1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:17_2"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#T15">WAYBEL_0:15</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:17_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 <b>and </b><br/><a NAME="E13:17_2"/>
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> )
 ;<br/><b>thus </b><a NAME="E14:17_2"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:17"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E12:17"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="funct_7.html#T70">FUNCT_7:70</a></i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>
<a NAME="E14:17"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 
;<br/>
<b>set </b><font color="Maroon">V</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_4"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:17_4"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E4:17_4"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/><b>thus </b><a NAME="E5:17_4"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">V</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E17:17"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/><font color="Olive">X</font> is <a href="waybel_0.html#V13">upper</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:17_5"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:17_5"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E4:17_5"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>
<b>thus </b><a NAME="E5:17_5"/>
<font color="Maroon">X</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:17"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/><font color="Olive">X</font> is <a href="waybel_0.html#V2">filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:17_6"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:17_6"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E4:17_6"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>
<b>thus </b><a NAME="E5:17_6"/>
<font color="Maroon">X</font> is <a href="waybel_0.html#V2">filtered</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E19:17"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">X</font>, <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> holds <br/> ex <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Olive">X</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:17_7"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">z1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:17_7"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font>
 <b>and </b><br/><a NAME="E4:17_7"/><i><font color="Green">E103</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 ;<br/>
<b>consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:17_7"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">z2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:17_7"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z2</font>
 <b>and </b><br/><a NAME="E9:17_7"/><i><font color="Green">E108</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:17_7"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">z2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">W</font> <b>by </b><i>, <a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/>
<a NAME="E13:17_7"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>;<br/>
<a NAME="E1:17_7_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
;<br/>
<a NAME="E2:17_7_1"/><i><font color="Green">E113</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:17_7_1_1"/><i><font color="Green">E114</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>
 ;<br/>

<b>set </b><font color="Maroon">nk</font> = <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>;<br/>
<a NAME="E2:17_7_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">znk</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:17_7_1_1"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">znk</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">znk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font></span>)</span> &amp; <font color="Maroon">znk</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:17_7_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><i><font color="Green">E118</font></i>: <font color="Maroon">znk</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>

;<br/>
<a NAME="E7:17_7_1_1"/>
<font color="Maroon">znk</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>
 
<b>by </b><i>, <a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a></i>;<br/>
<b>hence </b><a NAME="E8:17_7_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:17_7_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E4:17_7_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="waybel_6.html#K2" target="_self">.</a> <font color="Maroon">y1</font>
 ;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">z</font> = <font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font>;<br/>
<div><i><font color="Green">E119</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_7_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_7_2_1"/>
( <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font> or <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_2_1_1"><b>suppose </b></a><a NAME="E1:17_7_2_1_1"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:17_7_2_1_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">n1</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:17_7_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:17_7_2_1_1"/><b>then </b>
<font color="Maroon">z2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E6:17_7_2_1_1"/>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_7_2_1_2"><b>suppose </b></a><a NAME="E1:17_7_2_1_2"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:17_7_2_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">n2</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E4:17_7_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:17_7_2_1_2"/><b>then </b>
<font color="Maroon">z1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z2</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E6:17_7_2_1_2"/>
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T25">YELLOW_0:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E15:17_7"/>
( <font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z1</font> &amp; <font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z2</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23">YELLOW_0:23</a></i>;<br/>
<a NAME="E16:17_7"/><b>then </b>
(  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font></span>)</span> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z2</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_0.html#T22">WAYBEL_0:22</a></i>;<br/>
<a NAME="E17:17_7"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">z2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E18:17_7"/>
 ex <font color="Olive">Z</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">X</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:17_8"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_8"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:17_8"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:17_8"/><i><font color="Green">E124</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E7:17_8"/><i><font color="Green">E125</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:17_8"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">z1</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">z</font></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><a NAME="E11:17_8"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">z1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:17_8"/><b>then </b>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_0.html#T15">WAYBEL_0:15</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:17_8"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font>
 <b>and </b><br/><a NAME="E15:17_8"/><i><font color="Green">E128</font></i>: 
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> )
 ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E17:17_8"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">z1</font> )
 <b>by </b><i/>;<br/><a NAME="E18:17_8"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E19:17_8"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>then consider </b><font color="Maroon">z'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E21:17_8"/><i><font color="Green">E132</font></i>: 
( <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/><a NAME="E22:17_8"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">z'</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">u1</font>
 <b>by </b><i>, , , , <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><a NAME="E23:17_8"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 ;<br/><a NAME="E24:17_8"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E25:17_8"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>then </b><font color="Maroon">z'</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
;<br/><a NAME="E27:17_8"/><b>then </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">p</font></span>)</span> where <font color="Olive">p</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 ;<br/><a NAME="E28:17_8"/><b>then </b>
<font color="Maroon">z'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E29:17_8"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">g</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">u1</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as  <a href="waybel_6.html#V1" target="_self">Open</a> <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font> <b>by </b><i>, , , , , , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a>, <a class="ref" href="waybel_0.html#T28">WAYBEL_0:28</a>, <a class="ref" href="waybel_0.html#T47">WAYBEL_0:47</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_9"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:17_9"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:17_9"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">z</font></span>)</span> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:17_9"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E7:17_9"/><i><font color="Green">E140</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:17_9"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_6.html#K1" target="_self">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E11:17_9"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/><a NAME="E12:17_9"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E13:17_9"/><b>then </b>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/><a NAME="E14:17_9"/><b>then </b>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#T2">WAYBEL_3:2</a></i>;<br/><b>hence </b><a NAME="E15:17_9"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T8">WAYBEL_3:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E23:17"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
