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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">R</font> be  <a href="waybel35.html#V1" target="_self">extra-order</a> <a href="alg_1.html#NM1">Relation</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">C</font> be  <a href="waybel35.html#V3" target="_self">satisfying_SIC</a>  <a href="waybel35.html#M1" target="_self">strict_chain</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:52"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:52"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:52"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E4:52"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">R</font> <a href="waybel35.html#R1" target="_self">satisfies_SIC_on</a> <font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:52"/>
not <font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/>
<a NAME="E6:52"/><b>then </b>
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">q</font>,<font color="Maroon">p</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/>
<a NAME="E7:52"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, , , </i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:52"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E10:52"/><i><font color="Green">E61</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E11:52"/><i><font color="Green">E64</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">w</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E12:52"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, , , , </i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:52"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E15:52"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E16:52"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E17:52"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2"><i><font color="Green">Lemma26</font></i></a>, , <a class="txt" href="waybel35.html#E1:52"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel35.html#E2:52"><i><font color="Green">E31</font></i></a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> =  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">w</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="waybel35.html#T1" target="_self">Th1</a>, </i>;<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> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font> );<br/>
<a NAME="E19:52"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>



<a NAME="E1:52_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:52_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1"><b>suppose </b></a><a NAME="E1:52_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">w</font>
 ;<br/><div class="add"><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="E3:52_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:52_1_1_1"/><i><font color="Green">E74</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:52_1_1_1"/><i><font color="Green">E75</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E6:52_1_1_1"/>
<font color="Maroon">t</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="waybel35.html#E1:52"><i><font color="Green">E30</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">b</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="waybel35.html#D1" target="_self">Def1</a>, , </i>;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><b>take </b>
<font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E8:52_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="waybel35.html#D1" target="_self">Def1</a>, <a class="ref" href="waybel35.html#D2" target="_self">Def2</a>, , <a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">x</font>
;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><b>thus </b><a NAME="E2:52_1_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">t</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font> )
 <b>by </b><i><a class="ref" href="waybel35.html#D1" target="_self">Def1</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="waybel35.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">D</font><b> such that </b><br/><a NAME="E22:52"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E23:52"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="ref" href="waybel35.html#T3" target="_self">Th3</a>);<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>take </b><font color="Maroon">y</font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>;<br/>

<a NAME="E25:52"/><i><font color="Green">E79</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>,<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="E26:52"/><i><font color="Green">E80</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E27:52"/><b>then </b>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E28:52"/><b>then </b>
<font color="Maroon">x1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel35.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>
<b>hence </b><a NAME="E29:52"/>
<font color="Maroon">p</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel35.html#T2" target="_self">Th2</a>, <a class="ref" href="orders_2.html#T32">ORDERS_2:32</a></i>;<br/>

<a NAME="E30:52"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">q</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font>
 
;<br/>
<a NAME="E31:52"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:52_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E6:52_2"/><i><font color="Green">E110</font></i>: 
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="waybel35.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:52_2"/><b>then </b>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_4.html#D4">WAYBEL_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E8:52_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</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="E32:52"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="waybel35.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<b>hence </b><a NAME="E33:52"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:52"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="waybel_4.html#D5">WAYBEL_4:def 5</a></i>;<br/>

<a NAME="E34:52"/><i><font color="Green">E111</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">y</font>,<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="E35:52"/><i><font color="Green">E112</font></i>: 
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">y</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E36:52"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">y</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">x</font> holds <br/><font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_3"/><i><font color="Green">E113</font></i>: 
 <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">y</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:52_3"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</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="E1:52_3_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:52_3_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:52_3_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">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="txt" href="waybel35.html#E4:52"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E6:52_3_1"/><i><font color="Green">E116</font></i>: 
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="waybel35.html#D1" target="_self">Def1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>;<br/>
<a NAME="E7:52_3_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:52_3_1_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:52_3_1_1"/><i><font color="Green">E118</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">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="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_3_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:52_3_1_1_1"/>
<font color="Maroon">k</font> is    <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="E2:52_3_1_1_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</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"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">w</font> )
 
<b>by </b><i><a class="ref" href="waybel35.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:52_3_1_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1] )
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:52_3_1_1"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="waybel35.html#D3" target="_self">Def3</a>, );</i><br/>


</div><b>end;</b></div>
<a NAME="E8:52_3_1"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
;<br/>
<a NAME="E9:52_3_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
;<br/>
<a NAME="E10:52_3_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E11:52_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel35.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_4.html#T1">YELLOW_4:1</a></i>;<br/>
<a NAME="E12:52_3_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">m</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_4.html#D5">WAYBEL_4:def 5</a></i>;<br/>
<a NAME="E13:52_3_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="waybel35.html#D3" target="_self">Def3</a>, </i>;<br/>
<b>hence </b><a NAME="E14:52_3_1"/>
<font color="Maroon">m</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_3"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel35.html#D2" target="_self">Def2</a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E37:52"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" target="_self">SetBelow</a> <font color="Maroon">R</font>,<font color="Maroon">C</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


</div>
