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

<b>let </b><font color="Maroon">L</font> be  <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<a NAME="E1:6"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  ex <font color="Olive">a</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="lattice3.html#R3">is_less_than</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Olive">X</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><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>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">X'</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><br/> for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">X'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X'</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) );<br/>
<a NAME="E1:6_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:6_1_1"/>
for <font color="Olive">X'</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><br/> for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">X'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> 0 holds <br/> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X'</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X'</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E1:6_1_1_1"/><i><font color="Green">E31</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:6_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:6_1_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E2:6_1_1_1_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> holds  <font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E3:6_1_1_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E4:6_1_1_1_1"/>
for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:6_1_1_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6_1_1_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:6_1"/><i><font color="Green">E36</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>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</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="6_1_2"><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="E1:6_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:6_1_2"/>
for <font color="Olive">X'</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><br/> for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st  <a href="relat_1.html#K2">rng</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">X'</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 holds <br/> ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X'</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>assume </b><a NAME="E1:6_1_2_1"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">q</font> = <font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">X'</font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span>;<br/>
<a NAME="E2:6_1_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E3:6_1_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E4:6_1_2_1"/>
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>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_1"><b>proof </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="E1:6_1_2_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:6_1_2_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E3:6_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:6_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:6_1_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E7:6_1_2_1"/><i><font color="Green">E44</font></i>: 
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:6_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="funct_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> <font color="Maroon">b</font>
 ;<br/>
<a NAME="E10:6_1_2_1"/>
 <a href="finseq_1.html#K2">Seg</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>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E11:6_1_2_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E12:6_1_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">a2</font> = <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E14:6_1_2_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a></i>;<br/>
<a NAME="E15:6_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a></i>;<br/>
<a NAME="E16:6_1_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<i><font color="Green">E50</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green">E51</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E19:6_1_2_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_2_1_4"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1_2_1_4_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> or  not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_4_1_1"><b>suppose </b></a><a NAME="E1:6_1_2_1_4_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:6_1_2_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> )
 <b>by </b><i>, , , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/><b>hence </b><a NAME="E3:6_1_2_1_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2_1_4_1_2"><b>suppose </b></a><a NAME="E1:6_1_2_1_4_1_2"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:6_1_2_1_4_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E5:6_1_2_1_4_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:6_1_2_1_4_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E7:6_1_2_1_4_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_4_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1_2_1_4_1_2_1"/>
not <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:6_1_2_1_4_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="int_1.html#T20">INT_1:20</a></i>;<br/>
<a NAME="E3:6_1_2_1_4_1_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:6_1_2_1_4_1_2_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T9">FINSEQ_1:9</a></i>
;<br/>
<a NAME="E6:6_1_2_1_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="txt" href="lattice6.html#E1"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_2_1_4_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:6_1_2_1_4_1_2"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="txt" href="lattice6.html#E1:6_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E9:6_1_2_1_4_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E20:6_1_2_1"/>
for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_2_1_5"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 ;<br/>

<a NAME="E2:6_1_2_1_5"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> holds <br/><font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2_1_5_1"><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 <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<a NAME="E1:6_1_2_1_5_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<b>hence </b><a NAME="E2:6_1_2_1_5_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span></span>)</span> implies <font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_1_2_1_5"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:6_1_2_1_5"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , </i>;<br/>
<i><font color="Green">E66</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green">E68</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E8:6_1_2_1_5"/><b>then </b><i><font color="Green">E69</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , <a class="txt" href="lattice6.html#E1:6_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D7">LATTICES:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
<br/>.= 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/>
<a NAME="E10:6_1_2_1_5"/><b>then </b>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<b>hence </b><a NAME="E11:6_1_2_1_5"/>
<font color="Maroon">x'</font> <a href="orders_2.html#R3">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E1"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:6_1_2_1"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> st <br/>( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:6_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_1"/><i><font color="Green">E70</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>  holds  <font color="Maroon">S<sub>1</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>
<a NAME="E4:6_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E6:6_1"/>
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">L</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D2">LATTICE3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:6_1"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X'</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<a NAME="E10:6_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E12:6_1"/><i><font color="Green">E79</font></i>: 
( ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) &amp; ( for <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font> ) holds <br/><font color="Olive">x'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E13:6_1"/>
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">X</font> holds <br/> <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_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="E1:6_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span><b> such that </b><br/><a NAME="E3:6_1_3"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> )
 ;<br/>
<a NAME="E4:6_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E82</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E6:6_1_3"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:6_1_3"/>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:6_1"/><b>then </b><i><font color="Green">E83</font></i>: 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E15:6_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1_4"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E2:6_1_4"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font> holds <br/><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_4_1"><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 <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:6_1_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 ;<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="E3:6_1_4_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E4:6_1_4_1"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>
<a NAME="E5:6_1_4_1"/><b>then </b>
<font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x'</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E6:6_1_4_1"/>
<font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:6_1_4"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E5:6_1_4"/>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:6_1"/>
 ex <font color="Olive">a</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="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#R3">is_less_than</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:6"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V4">complete</a>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D6">VECTSP_8:def 6</a></i>;<br/>


</div>
