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<b>consider </b><font color="Maroon">G</font> being  <a href="glib_000.html#V2">finite</a> <a href="glib_002.html#V3" target="_self">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>
<b>set </b><font color="Maroon">F</font> = <a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">G</font>;<br/>
<a NAME="E2:58_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K1">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">G</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">F</font> = <a href="numbers.html#K5">NAT</a>  <a href="pboole.html#K9">--&gt;</a> <font color="Maroon">G</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="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:58_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">G</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="E3:58_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">F</font> = <font color="Maroon">F</font> as   <a href="glib_000.html#NM11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55">GLIB_000:def 55</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<a NAME="E6:58_1_1"/>
( <font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="glib_000.html#K42">.</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">F</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:58_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">G</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">F</font> <a href="glib_000.html#K42">.</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">G</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="E8:58_1_1"/><b>then </b>
( <font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">F</font> <a href="glib_000.html#K42">.</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">G</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:58_1_1"/>
<font color="Maroon">F</font> is <a href="glib_000.html#V13">halting</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D56">GLIB_000:def 56</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:58_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:58_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:58_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">G</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="E4:58_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:58_1_1_2"/>
( <font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">x</font> is <a href="glib_000.html#V2">finite</a> &amp; <font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> <font color="Maroon">x</font> is <a href="glib_002.html#V3" target="_self">Tree-like</a> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:58_1_1"/>
( <font color="Maroon">F</font> is <a href="glib_000.html#V14">finite</a> &amp; <font color="Maroon">F</font> is <a href="glib_002.html#V8" target="_self">Tree-like</a> )
 <b>by </b><i><a class="ref" href="glib_000.html#D60">GLIB_000:def 60</a>, </i>;<br/>


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