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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  <a href="glib_000.html#V2" target="_self">finite</a> <a href="glib_000.html#V3" target="_self">loopless</a> <a href="glib_000.html#V4" target="_self">trivial</a> <a href="glib_000.html#V5" target="_self">non-multi</a> <a href="glib_000.html#V6" target="_self">non-Dmulti</a> <a href="glib_000.html#V7" target="_self">simple</a> <a href="glib_000.html#V8" target="_self">Dsimple</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E2:93_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">c<sub>1</sub></font> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:93_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:93_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E2:93_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><b>hence </b><a NAME="E3:93_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is   <a href="glib_000.html#NM2">_Graph</a>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="glib_000.html#NM11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55" target="_self">Def55</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<a NAME="E6:93_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="glib_000.html#E2:93_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:93_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E8:93_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:93_1_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V13" target="_self">halting</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D56" target="_self">Def56</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:93_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="glib_000.html#E2:93_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E2:93_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/><a NAME="E3:93_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:93_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V2" target="_self">finite</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V3" target="_self">loopless</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V4" target="_self">trivial</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V5" target="_self">non-multi</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V6" target="_self">non-Dmulti</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V7" target="_self">simple</a> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="glib_000.html#K42" target="_self">.-&gt;</a> <font color="Maroon">c<sub>5</sub></font> is <a href="glib_000.html#V8" target="_self">Dsimple</a> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:93_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V14" target="_self">finite</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V15" target="_self">loopless</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V16" target="_self">trivial</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V18" target="_self">non-multi</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V19" target="_self">non-Dmulti</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V20" target="_self">simple</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="glib_000.html#V21" target="_self">Dsimple</a> )
 <b>by </b><i><a class="ref" href="glib_000.html#D60" target="_self">Def60</a>, <a class="ref" href="glib_000.html#D61" target="_self">Def61</a>, <a class="ref" href="glib_000.html#D62" target="_self">Def62</a>, <a class="ref" href="glib_000.html#D64" target="_self">Def64</a>, <a class="ref" href="glib_000.html#D65" target="_self">Def65</a>, <a class="ref" href="glib_000.html#D66" target="_self">Def66</a>, <a class="ref" href="glib_000.html#D67" target="_self">Def67</a></i>;<br/>


</div>
