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

<b>consider </b><font color="Maroon" title="c1">G</font> being  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/>
<b>set </b><font color="Maroon" title="c2">F</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E2:92_1_1"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">F</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">G</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a></span><br/><a NAME="E1:92_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:92_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="#E2:92_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:92_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><b>hence </b><a NAME="E4:92_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">F</font> as   <a href="glib_000.html#NM11" title="GLIB_000:NM.11">GraphSeq</a> <b>by </b><i><a class="ref" href="glib_000.html#D55" target="_self" title="GLIB_000:def.55">Def55</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V15" title="GLIB_000:attr.15">loopless</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V16" title="GLIB_000:attr.16">trivial</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V18" title="GLIB_000:attr.18">non-multi</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V19" title="GLIB_000:attr.19">non-Dmulti</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V20" title="GLIB_000:attr.20">simple</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V21" title="GLIB_000:attr.21">Dsimple</a> )</span><br/>

<a NAME="E6:92_1_1"/>
( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> )
 
<b>by </b><i><a class="txt" href="#E2:92_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:92_1_1"/><b>then </b>
( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E8:92_1_1"/><b>then </b>
( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E9:92_1_1"/>
<font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D56" target="_self" title="GLIB_000:def.56">Def56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V15" title="GLIB_000:attr.15">loopless</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V16" title="GLIB_000:attr.16">trivial</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V18" title="GLIB_000:attr.18">non-multi</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V19" title="GLIB_000:attr.19">non-Dmulti</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V20" title="GLIB_000:attr.20">simple</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V21" title="GLIB_000:attr.21">Dsimple</a> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a> )</span><br/><a NAME="E1:92_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E2:92_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="txt" href="#E2:92_1_1"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:92_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">G</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><b>hence </b><a NAME="E4:92_1_1_2"/>
( <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V3" title="GLIB_000:attr.3">loopless</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V6" title="GLIB_000:attr.6">non-Dmulti</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V7" title="GLIB_000:attr.7">simple</a> &amp; <font color="Maroon" title="c4">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Maroon" title="c5">x</font> is <a href="glib_000.html#V8" title="GLIB_000:attr.8">Dsimple</a> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:92_1_1"/>
( <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V15" title="GLIB_000:attr.15">loopless</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V16" title="GLIB_000:attr.16">trivial</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V18" title="GLIB_000:attr.18">non-multi</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V19" title="GLIB_000:attr.19">non-Dmulti</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V20" title="GLIB_000:attr.20">simple</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="glib_000.html#V21" title="GLIB_000:attr.21">Dsimple</a> )
 <b>by </b><i><a class="ref" href="glib_000.html#D60" target="_self" title="GLIB_000:def.60">Def60</a>, <a class="ref" href="glib_000.html#D61" target="_self" title="GLIB_000:def.61">Def61</a>, <a class="ref" href="glib_000.html#D62" target="_self" title="GLIB_000:def.62">Def62</a>, <a class="ref" href="glib_000.html#D64" target="_self" title="GLIB_000:def.64">Def64</a>, <a class="ref" href="glib_000.html#D65" target="_self" title="GLIB_000:def.65">Def65</a>, <a class="ref" href="glib_000.html#D66" target="_self" title="GLIB_000:def.66">Def66</a>, <a class="ref" href="glib_000.html#D67" target="_self" title="GLIB_000:def.67">Def67</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
