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

<b>consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">E</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">S</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">T</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">E</font>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">Ga</font> =  <a href="glib_000.html#K12">createGraph</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>
<b>set </b><font color="Maroon">vl</font> = <font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1;<br/>
<a NAME="E2:110"/><i><font color="Green">E34</font></i>: 
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#T8">GLIB_000:8</a></i>;<br/>
<a NAME="E3:110"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E4:110"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="glib_000.html#T8">GLIB_000:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">vl</font> = <font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> 1 as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>, <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">Gb</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font>;<br/>
<b>set </b><font color="Maroon">G0</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> ;<br/>
<b>set </b><font color="Maroon">Fa</font> = <a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span>;<br/>
<a NAME="E6:110"/><i><font color="Green">E36</font></i>: 
(  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E18:110"><i><font color="Green">E46</font></i></a>, <a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>;<br/>
<a NAME="E7:110"/><i><font color="Green">E38</font></i>: 
(  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vl</font> &amp;  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/>
<a NAME="E8:110"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<a NAME="E9:110"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">vl</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E10:110"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span> is <a href="seqm_3.html#V6">natural-yielding</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green">E48</font></i></a>, <a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>
<a NAME="E11:110"/><i><font color="Green">E42</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span> is <a href="seqm_3.html#V6">natural-yielding</a>
 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/>
<a NAME="E12:110"/><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> is <a href="lexbfs.html#V3" target="_self">natural-vlabeled</a> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a>  is <a href="lexbfs.html#V3" target="_self">natural-vlabeled</a> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:110"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></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/>
<a NAME="E14:110"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E15:110"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <a href="numbers.html#K5">NAT</a>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E16:110"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> 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/>
<a NAME="E18:110"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> &amp; 0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/>
<a NAME="E19:110"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
 
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/>
<a NAME="E20:110"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/>
<a NAME="E21:110"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:110_1"/><i><font color="Green">E50</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:110_1"/>
not <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:110_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K5">NAT</a>  <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/>
<a NAME="E4:110_1"/>
<font color="Maroon">n</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/>
<b>hence </b><a NAME="E5:110_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_2_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_1"><b>suppose </b></a><a NAME="E1:110_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:110_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:110_2_1_1"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2"><b>suppose </b></a><a NAME="E1:110_2_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:110_2_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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/><b>hence </b><a NAME="E3:110_2_1_2"/>
( <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E23:110"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> is   <a href="glib_000.html#NM2">_Graph</a>
 
<b>by </b><i/>;<br/>
<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/>
<a NAME="E25:110"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="glib_000.html#K42">.</a> <font color="Olive">n</font> is <a href="glib_003.html#V3">[VLabeled]</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="glib_003.html#NM31">VGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D26">GLIB_003:def 26</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<div><i><font color="Green">E90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_3_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_3_1_1"><b>suppose </b></a><a NAME="E1:110_3_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:110_3_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E3:110_3_1_1"/>
( <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_3_1_2"><b>suppose </b></a><a NAME="E1:110_3_1_2"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:110_3_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <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/><b>hence </b><a NAME="E3:110_3_1_2"/>
( <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> )
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E28:110"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> &amp; <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font> )
 
<b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E29:110"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E30:110"/><b>then </b><i><font color="Green">E93</font></i>: 
<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="110_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:110_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_4_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_4_1_1"><b>suppose </b></a><a NAME="E1:110_4_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:110_4_1_1"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><a NAME="E3:110_4_1_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E4:110_4_1_1"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E5:110_4_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a>, <a class="ref" href="lexbfs.html#D1" target="_self">Def1</a>, , <a class="txt" href="lexbfs.html#E21:110"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E6:110_4_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>hence </b><a NAME="E7:110_4_1_1"/>
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_4_1_2"><b>suppose </b></a><a NAME="E1:110_4_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:110_4_1_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E3:110_4_1_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12">createGraph</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">E</font>,<font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<font color="Maroon">vl</font>
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:110_4_1_2"/>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:110_4_1_2"/>
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="lexbfs.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E32:110"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">F</font> is <a href="lexbfs.html#V9" target="_self">iterative</a>
 <b>by </b><i/>;<br/>

<div><i><font color="Green">E99</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:110_5"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>assume </b><a NAME="E2:110_5"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E3:110_5"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E4:110_5"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:110_5"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E6:110_5"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="lexbfs.html#T2" target="_self">Th2</a>, <a class="txt" href="lexbfs.html#E21:110"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E34:110"/>
<font color="Maroon">F</font> is <a href="lexbfs.html#V10" target="_self">eventually-constant</a>
 <b>by </b><i>, <a class="ref" href="lexbfs.html#D1" target="_self">Def1</a>, </i>;<br/>

<a NAME="E35:110"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Olive">n</font> is <a href="glib_000.html#V2">finite</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E36:110"/>
<font color="Maroon">F</font> is <a href="glib_000.html#V14">finite</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D60">GLIB_000:def 60</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:110_6"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">v</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E19:110"><i><font color="Green">E47</font></i></a>, </i>;<br/><b>hence </b><a NAME="E2:110_6"/>
 <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E19:110"><i><font color="Green">E47</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E38:110"/>
<font color="Maroon">F</font> is <a href="lexbfs.html#V13" target="_self">fixed-vertices</a>
 <b>by </b><i/>;<br/>

<a NAME="E39:110"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Olive">n</font> is <a href="lexbfs.html#V3" target="_self">natural-vlabeled</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2"><i><font color="Green">Lemma52</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E40:110"/>
<font color="Maroon">F</font> is <a href="lexbfs.html#V11" target="_self">natural-vlabeled</a>
 <b>by </b><i/>;<br/>

<a NAME="E41:110"/>
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a>, </i>;<br/>
<b>hence </b><a NAME="E42:110"/>
<font color="Maroon">F</font> is <a href="lexbfs.html#V18" target="_self">vlabel-initially-empty</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E11:110"><i><font color="Green">E42</font></i></a></i>;<br/>

<a NAME="E43:110"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F</font> <a href="glib_000.html#K40">.Lifespan()</a>  holds <br/> ex <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>  st <br/>( not <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Olive">n</font></span>)</span></span>)</span> &amp;  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Olive">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Olive">w</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Olive">n</font></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:110_7"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">F</font> <a href="glib_000.html#K40">.Lifespan()</a> 
 ;<br/>

<a NAME="E2:110_7"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a>, </i>;<br/>
<a NAME="E3:110_7"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:110_7"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>take </b>
<font color="Maroon">v</font>
;<br/>

<a NAME="E5:110_7"/><i><font color="Green">E103</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K6">-</a> 0
 
<b>by </b><i>, , <a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_000.html#D57">GLIB_000:def 57</a></i>;<br/>
<a NAME="E6:110_7"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="lexbfs.html#D2" target="_self">Def2</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:110_7"/>
( not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="lexbfs.html#K8" target="_self">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="glib_000.html#K40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green">E48</font></i></a>, <a class="ref" href="lexbfs.html#T3" target="_self">Th3</a>, , , <a class="ref" href="funct_4.html#T21">FUNCT_4:21</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E44:110"/>
<font color="Maroon">F</font> is <a href="lexbfs.html#V19" target="_self">adds-one-at-a-step</a>
 <b>by </b><i/>;<br/>


</div>
