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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:35_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> is <a href="waybel_3.html#V1" target="_self">compact</a>
 ;<br/><b>given </b><font color="Maroon">Y</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E2:35_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/> ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">y</font> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>2</sub></font> );<br/><div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<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="E4:35_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">z</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">u</font> = <font color="Maroon">u</font>;<br/><a NAME="E6:35_1_1"/>
( <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">z</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><b>hence </b><a NAME="E7:35_1_1"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">u</font>] )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:35_1"/><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Y</font>;<br/><b>set </b><font color="Maroon">x1</font> = <font color="Maroon">x</font>;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> ;<br/><a NAME="E7:35_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:35_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">x'</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E10:35_1"/><i><font color="Green">E44</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/><a NAME="E11:35_1"/><b>then </b>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,1</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> )
 <b>by </b><i><a class="ref" href="funct_7.html#T72">FUNCT_7:72</a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E12:35_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  &amp; <font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x'</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/><a NAME="E13:35_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:35_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/>
<a NAME="E4:35_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T76">FUNCT_7:76</a></i>;<br/>
<a NAME="E5:35_1_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_7.html#T76">FUNCT_7:76</a></i>;<br/>
<b>hence </b><a NAME="E6:35_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> where <font color="Olive">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : verum </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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">z</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> );<br/><a NAME="E1:35_1_3"/>
<span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:35_1_3"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 ;<br/><a NAME="E3:35_1_3"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_3_1"><b>proof </b></a><div class="add">

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

<b>given </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:35_1_3_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<b>take </b>
<font color="Maroon">z</font>
;<br/>

<a NAME="E2:35_1_3_1"/><i><font color="Green">E58</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</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">Y</font> )
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T76">FUNCT_7:76</a></i>;<br/>
<a NAME="E3:35_1_3_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:35_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:35_1_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">yy</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>take </b>
<font color="Maroon">yy</font>
;<br/>

<b>thus </b><a NAME="E7:35_1_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>

<a NAME="E8:35_1_3_1"/>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>;<br/>
<b>hence </b><a NAME="E9:35_1_3_1"/>
<font color="Maroon">yy</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</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="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>

<a NAME="E10:35_1_3_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">yy</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:35_1_3_1"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yy</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E12:35_1_3_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yy</font>
 <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="E4:35_1_3"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  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>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:35_1_3"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:35_1_3"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E9:35_1_3"/><i><font color="Green">E89</font></i>: 
 ex <font color="Olive">z</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">y</font> )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">z</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="E10:35_1_3"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 ;<br/><b>hence </b><a NAME="E11:35_1_3"/>
<font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_4"><b>now </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>, <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="E1:35_1_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:35_1_4"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><b>assume </b><a NAME="E4:35_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>then consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:35_1_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">k2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E7:35_1_4"/>
( <font color="Maroon">k1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k2</font> or <font color="Maroon">k2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k1</font> )
 ;<br/><b>hence </b><a NAME="E8:35_1_4"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> or <font color="Maroon">z</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E17:35_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> is <a href="waybel_3.html#V1" target="_self">compact</a>
 <b>by </b><i/>;<br/><a NAME="E18:35_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E20:35_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_5"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:35_1_5"/><b>then </b>
<font color="Maroon">x</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T6">YELLOW_0:6</a></i>;<br/><a NAME="E3:35_1_5"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E4:35_1_5"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">Z</font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E5:35_1_5"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/><b>hence </b><a NAME="E6:35_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/></div><b>end;</b></div><a NAME="E22:35_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E23:35_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 ;<br/><div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:35_1_6"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><b>assume </b><a NAME="E3:35_1_6"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B</font>]
 ;<br/><b>thus </b><a NAME="E4:35_1_6"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_6_1"/>
<font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:35_1_6_1"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">B</font>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E3:35_1_6_1"/><b>then </b><i><font color="Green">E102</font></i>: 
(  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font></span>)</span> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x'</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T36">YELLOW_0:36</a>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_6_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_6_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:35_1_6_1_1_1"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_6_1_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:35_1_6_1_1_2"/><b>then </b>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:35_1_6_1_1_2"/><b>then </b>
( <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font> or <font color="Maroon">x'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E4:35_1_6_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font> or ( <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font></span>)</span> &amp; <font color="Maroon">x'</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">B</font>,<font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> ) )
 <b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/><a NAME="E5:35_1_6_1_1_2"/><b>then </b>
(  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:35_1_6_1_1_2"/>
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><a NAME="E25:35_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(, , <a class="ref" href="waybel_3.html#D1" target="_self">Def1</a>);<br/></i><a NAME="E26:35_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E28:35_1"/><i><font color="Green">E104</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E29:35_1"/><i><font color="Green">E105</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>then reconsider </b><font color="Maroon">xSn</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E31:35_1"/>
(  <a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="funct_7.html#T73">FUNCT_7:73</a>, <a class="ref" href="funct_7.html#T76">FUNCT_7:76</a></i>;<br/><a NAME="E32:35_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">xSn</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9">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">x</font> )
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E33:35_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">A</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">xSn</font> &amp; <font color="Maroon">xSn</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#D2" target="_self">Def2</a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a>, <a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/><a NAME="E34:35_1"/><b>then </b>
( <font color="Maroon">xSn</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">Z</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">xSn</font> )
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/><b>hence </b><a NAME="E35:35_1"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:35"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">X</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/> not <font color="Olive">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">y</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D1" target="_self">WAYBEL_3:def 1</a>,<a class="ref" href="waybel_3.html#D2" target="_self">WAYBEL_3:def 2</a><br/>



<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:35"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; ( 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">D</font> holds <br/> not <font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Olive">z</font> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:35"/>
<font color="Maroon">D</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then 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="E3:35_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a></i>;<br/>
<a NAME="E4:35_2"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:35_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:35"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E7:35"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E8:35"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E9:35"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>
<b>hence </b><a NAME="E10:35"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> implies  ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> ) )
 <b>by </b><i/>;<br/>


</div>
