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

<b>set </b><font color="Maroon">VV</font> =  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">V1</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>;<br/>
<b>set </b><font color="Maroon">V2</font> =  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">EE</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> ) </span>}</span> ;<br/>
<a NAME="E1:45_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</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="E2:45_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
(  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp;  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<a NAME="E3:45_1_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j0</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i0</font>,<font color="Maroon">j0</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">i0</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Maroon">j0</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/>
<a NAME="E4:45_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">i0</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j0</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">i0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j0</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i0</font>,<font color="Maroon">j0</font></span><a href="tarski.html#K2">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1_1_1_1"/>
( <font color="Maroon">i0</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j0</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:45_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="sgraph1.html#T6" target="_self">Th6</a></i>;<br/>

<a NAME="E2:45_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:45_1_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="xboole_0.html#R1">misses</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:45_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span> <a href="finsub_1.html#K3">/\</a> <span class="p1">(<span class="default"><a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:45_1_1_1_1"/>
( <font color="Maroon">i0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j0</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">i0</font>,<font color="Maroon">j0</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="sgraph1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:45_1_1_1"/>
<font color="Maroon">e</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:45_1_1_1_2"/>
for <font color="Olive">e0</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e</font> holds <br/><font color="Olive">e0</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">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:45_1_1_1_2"/>
<font color="Maroon">e</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:45_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:45_1_1"/>
 <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">EE</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="sgraph1.html#T5" target="_self">Th5</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>set </b><font color="Maroon">IT</font> =  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">EE</font> #);<br/>
<a NAME="E6:45_1_1"/>
 <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">EE</font> #) <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K3" target="_self">SIMPLEGRAPHS</a> <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">IT</font> =  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">EE</font> #) as    <a href="sgraph1.html#M1" target="_self">SimpleGraph</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">EE</font> = <font color="Maroon">EE</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">IT</font>
;<br/>

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

<b>thus </b><a NAME="E9:45_1_1"/>
( <font color="Maroon">EE</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K2">}</a></span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Olive">j</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K1" target="_self">nat_interval</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> ) </span>}</span>  &amp; <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="sgraph1.html#G1" target="_self">SimpleGraphStruct</a>(# <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">EE</font> #) )
 ;<br/>


</div>
