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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; not <font color="Olive">b<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> holds <br/>ex <font color="Olive">b<sub>4</sub></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="Olive">b<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>5</sub></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="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>4</sub></font>;<br/>
<a NAME="E1:4"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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">c<sub>3</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E5</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<a NAME="E2:4_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></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/>
<b>hence </b><a NAME="E3:4_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a> or ex <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:4"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_1"><b>suppose </b></a><a NAME="E1:4_2_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:4_2_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 holds <br/>ex <font color="Olive">b<sub>4</sub></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="Olive">b<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>5</sub></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="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <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">c<sub>4</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E1:4_2_1_1_1"/><i><font color="Green">E8</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<a NAME="E2:4_2_1_1_1"/><b>then </b><i><font color="Green">E9</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:4_2_1_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>
<a NAME="E5:4_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:4_2_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:4_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:4_2_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></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">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_1_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E8:4_2_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:4_2_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="lattice6.html#E4:4_2_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>;<br/>
<a NAME="E9:4_2_1_1_1"/><b>then </b>
for <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:4_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:4_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_1"><i><font color="Green">E7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_2"><b>suppose </b></a><a NAME="E1:4_2_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:4_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Olive">b<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 holds <br/>ex <font color="Olive">b<sub>4</sub></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="Olive">b<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>5</sub></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="Olive">b<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>5</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="group_1.html#V6">finite</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <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">c<sub>4</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>





<b>assume </b><a NAME="E1:4_2_1_2_1"/><i><font color="Green">E8</font></i>: 
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E2:4_2_1_2_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></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="E3:4_2_1_2_1"/><b>then </b><i><font color="Green">E10</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E4:4_2_1_2_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:4_2_1_2_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:4_2_1_2_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E3:4_2_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:4_2_1_2_1_1"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:4_2_1_2_1"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:4_2_1_2_1"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="lattice6.html#E3:4_2_1_2_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E7:4_2_1_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E9:4_2_1_2_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="lattice6.html#E3:4_2_1_2_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="lattice6.html#E5:4_2_1_2_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:4_2_1_2_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>
<a NAME="E12:4_2_1_2_1"/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:4_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E14:4_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E11:4_2_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></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">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>;<br/>
<i><font color="Green">E14</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></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">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T17">LATTICES:17</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></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">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#T17">LATTICES:17</a></i>
;<br/>
<a NAME="E18:4_2_1_2_1"/><b>then </b><i><font color="Green">E15</font></i>: 
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E16:4_2_1_2_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<i><font color="Green">E16</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green">E17</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<a NAME="E21:4_2_1_2_1"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></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">c<sub>4</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:4_2_1_2_1_6"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_2_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_2_1_2_1_6_1"/>
( <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> or  not <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_2_1_2_1_6_1_1"><b>suppose </b></a><a NAME="E1:4_2_1_2_1_6_1_1"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:4_2_1_2_1_6_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E9:4_2_1_2_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="lattice6.html#E18:4_2_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lattice6.html#E19:4_2_1_2_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/><b>hence </b><a NAME="E3:4_2_1_2_1_6_1_1"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> 
 <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="4_2_1_2_1_6_1_2"><b>suppose </b></a><a NAME="E1:4_2_1_2_1_6_1_2"/><i><font color="Green">E19</font></i>: 
not <font color="Maroon">c<sub>13</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2_1_2_1_6_1_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="lattice6.html#E1:4_2_1_2_1_6"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="lattice6.html#E3:4_2_1_2_1_6_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E5:4_2_1_2_1_6_1_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>15</sub></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">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="lattice6.html#E3:4_2_1_2_1_6_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E6:4_2_1_2_1_6_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></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="E7:4_2_1_2_1_6_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2_1_2_1_6_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_2_1_2_1_6_1_2_1"/>
not <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E2:4_2_1_2_1_6_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice6.html#E5:4_2_1_2_1_6_1_2"><i><font color="Green">E21</font></i></a>, <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="E3:4_2_1_2_1_6_1_2_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:4_2_1_2_1_6_1_2_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>15</sub></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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="lattice6.html#E2:4_2_1_2_1"><i><font color="Green">E9</font></i></a>, <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">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></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">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1"><i><font color="Green">E8</font></i></a>, <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">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E2:4_2_1_2_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T9">FINSEQ_1:9</a></i>
;<br/>
<a NAME="E6:4_2_1_2_1_6_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:4_2_1_2_1_6_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lattice6.html#E3:4_2_1_2_1_6_1_2_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E7:4_2_1_2_1_6_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice6.html#E1:4_2_1_2_1_6_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="lattice6.html#E4:4_2_1_2_1_6_1_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:4_2_1_2_1_6_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E11:4_2_1_2_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="lattice6.html#E3:4_2_1_2_1_6_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lattice6.html#E6:4_2_1_2_1_6_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E9:4_2_1_2_1_6_1_2"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>9</sub></font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E18:4_2_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lattice6.html#E20:4_2_1_2_1"><i><font color="Green">E17</font></i></a>, <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>
<b>hence </b><a NAME="E22:4_2_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></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">c<sub>4</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:4_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:4"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="lattice6.html#E1:4"><i><font color="Green">E4</font></i></a>, <a class="txt" href="lattice6.html#E2:4"><i><font color="Green">E5</font></i></a>);<br/></i>
<a NAME="E4:4"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E6:4"/>
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <font color="Maroon">c<sub>1</sub></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">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E9:4"/><i><font color="Green">E7</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T73">FINSEQ_1:73</a></i>;<br/>
<a NAME="E10:4"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E11:4"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="xboole_0.html#V1">empty</a> or ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></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">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E3:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="lattice6.html#E9:4"><i><font color="Green">E7</font></i></a></i>;<br/>


</div>
