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

<b>consider </b><font color="Maroon" title="c1">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c2">E</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">T</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">E</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c5">Ga</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font>;<br/>
<b>set </b><font color="Maroon" title="c6">vl</font> = <font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 1;<br/>
<a NAME="E2:110"/><i><font color="Green" title="E46">A1</font></i>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="glib_000.html#T8" title="GLIB_000:th.8">GLIB_000:8</a></i>;<br/>
<a NAME="E3:110"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E4:110"/><b>then </b>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 1</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="glib_000.html#T8" title="GLIB_000:th.8">GLIB_000:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">vl</font> = <font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> 1 as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">Gb</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font>;<br/>
<b>set </b><font color="Maroon" title="c9">G0</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<b>set </b><font color="Maroon" title="c10">Fa</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">F</font> = <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>;<br/>
<a NAME="E6:110"/><i><font color="Green" title="E47">A2</font></i>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E2:110"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="glib_000.html#T12" title="GLIB_000:th.12">GLIB_000:12</a></i>;<br/>
<a NAME="E7:110"/><i><font color="Green" title="E48">A3</font></i>: 
(  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">vl</font> &amp;  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></i>;<br/>
<a NAME="E8:110"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">vl</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<a NAME="E9:110"/><b>then </b><i><font color="Green" title="E49">A4</font></i>: 
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span> is <a href="seqm_3.html#V6" title="SEQM_3:attr.6">natural-yielding</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E7:110"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="seqm_3.html#D8" title="SEQM_3:def.8">SEQM_3:def 8</a></i>;<br/>
<a NAME="E10:110"/>
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> is <a href="seqm_3.html#V6" title="SEQM_3:attr.6">natural-yielding</a>
 
<b>by </b><i><a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></i>;<br/>
<a NAME="E11:110"/><b>then </b><i><font color="Green" title="E50">A5</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E9:110"><i><font color="Green" title="E49">A4</font></i></a>, <a class="ref" href="lexbfs.html#D7" target="_self" title="LEXBFS:def.7">Def7</a></i>;<br/>
<a NAME="E12:110"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></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/>
<a NAME="E13:110"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/>
<a NAME="E14:110"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E15:110"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span></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="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">F</font> = <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span> 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/>
<a NAME="E17:110"/><i><font color="Green" title="E51">A6</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:110"/><b>then </b><i><font color="Green" title="E52">A7</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/>
<a NAME="E19:110"/><b>then </b><i><font color="Green" title="E53">A8</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/>
<a NAME="E20:110"/><i><font color="Green" title="E54">A9</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">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" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:110_1"/><i><font color="Green" title="E55">A10</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">n</font>
 ;<br/>

<a NAME="E2:110_1"/>
not <font color="Maroon" title="c13">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E17:110"><i><font color="Green" title="E51">A6</font></i></a>, <a class="txt" href="lexbfs.html#E1:110_1"><i><font color="Green" title="E55">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:110_1"/><b>then </b><i><font color="Green" title="E56">A11</font></i>: 
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font>
 
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/>
<a NAME="E4:110_1"/>
<font color="Maroon" title="c13">n</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/>
<b>hence </b><a NAME="E5:110_1"/>
<font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E3:110_1"><i><font color="Green" title="E56">A11</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E55">A12</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" title="c13">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="c13">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:110_2_1_1"/><b>then </b>
<font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E3:110_2_1_1"/>
( <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E18:110"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</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" title="c13">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:110_2_1_2"/><b>then </b>
<font color="Maroon" title="c13">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E3:110_2_1_2"/>
( <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green" title="E54">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E22:110"/><b>then </b>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">F</font> = <font color="Maroon" title="c12">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" title="GLIB_000:def.55">GLIB_000:def 55</a></i>;<br/>
<a NAME="E24:110"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c13">F</font> <a href="glib_000.html#K42" title="GLIB_000:func.42">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E21:110"><i><font color="Green" title="E55">A12</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">F</font> = <font color="Maroon" title="c13">F</font> as   <a href="glib_003.html#NM31" title="GLIB_003:NM.31">VGraphSeq</a> <b>by </b><i><a class="ref" href="glib_003.html#D26" title="GLIB_003:def.26">GLIB_003:def 26</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">F</font>
;<br/>

<div><i><font color="Green" title="E56">A13</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" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </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" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:110_3_1_1"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E3:110_3_1_1"/>
( <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E18:110"><i><font color="Green" title="E52">A7</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</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" title="c15">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:110_3_1_2"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E3:110_3_1_2"/>
( <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green" title="E54">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E27:110"/><i><font color="Green" title="E57">A14</font></i>: 
( <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> &amp; <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font> )
 
<b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green" title="E54">A9</font></i></a></i>;<br/>
<a NAME="E28:110"/><b>then </b><i><font color="Green" title="E58">A15</font></i>: 
<font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E29:110"/><b>then </b><i><font color="Green" title="E59">A16</font></i>: 
<font color="Maroon" title="c14">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" title="GLIB_000:def.56">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" title="c15">k</font>, <font color="Maroon" title="c16">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>assume </b><a NAME="E1:110_4"/>
<font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c16">n</font>
 ;<br/><a NAME="E2:110_4"/>
<font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E3:110_4"/><b>then </b><i><font color="Green" title="E60">A20</font></i>: 
<font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font></span>)</span> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K4" title="GLIB_003:func.4">VLabelSelector</a> ,<font color="Maroon" title="c7">vl</font>
 <b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green" title="E54">A9</font></i></a></i>;<br/><a NAME="E4:110_4"/>
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E5:110_4"/>
<font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E20:110"><i><font color="Green" title="E54">A9</font></i></a>, <a class="txt" href="lexbfs.html#E3:110_4"><i><font color="Green" title="E60">A20</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E31:110"/><i><font color="Green" title="E60">A21</font></i>: 
<font color="Maroon" title="c14">F</font> is <a href="lexbfs.html#V9" title="LEXBFS:attr.9">iterative</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D15" target="_self" title="LEXBFS:def.15">Def15</a></i>;<br/>

<div><i><font color="Green" title="E61">A22</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" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:110_5"/><i><font color="Green" title="E62">A23</font></i>: 
<font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><b>assume </b><a NAME="E2:110_5"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ;<br/><a NAME="E3:110_5"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E4:110_5"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> &amp; <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:110_5"/><b>then </b>
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>hence </b><a NAME="E6:110_5"/>
contradiction
 <b>by </b><i><a class="txt" href="lexbfs.html#E7:110"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="lexbfs.html#E19:110"><i><font color="Green" title="E53">A8</font></i></a>, <a class="txt" href="lexbfs.html#E20:110"><i><font color="Green" title="E54">A9</font></i></a>, <a class="txt" href="lexbfs.html#E1:110_5"><i><font color="Green" title="E62">A23</font></i></a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E33:110"/>
<font color="Maroon" title="c14">F</font> is <a href="lexbfs.html#V10" title="LEXBFS:attr.10">eventually-constant</a>
 <b>by </b><i><a class="txt" href="lexbfs.html#E29:110"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="lexbfs.html#E31:110"><i><font color="Green" title="E60">A21</font></i></a>, <a class="ref" href="lexbfs.html#T14" target="_self" title="LEXBFS:th.14">Th14</a></i>;<br/>

<a NAME="E34:110"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Olive" title="b1">n</font> is <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E26:110"><i><font color="Green" title="E56">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E35:110"/>
<font color="Maroon" title="c14">F</font> is <a href="glib_000.html#V14" title="GLIB_000:attr.14">finite</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D60" title="GLIB_000:def.60">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" title="c15">n</font>, <font color="Maroon" title="c16">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:110_6"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E6:110"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="lexbfs.html#E26:110"><i><font color="Green" title="E56">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E2:110_6"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c16">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="lexbfs.html#E6:110"><i><font color="Green" title="E47">A2</font></i></a>, <a class="txt" href="lexbfs.html#E26:110"><i><font color="Green" title="E56">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E37:110"/>
<font color="Maroon" title="c14">F</font> is <a href="lexbfs.html#V13" title="LEXBFS:attr.13">fixed-vertices</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D19" target="_self" title="LEXBFS:def.19">Def19</a></i>;<br/>

<a NAME="E38:110"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Olive" title="b1">n</font> is <a href="lexbfs.html#V3" title="LEXBFS:attr.3">natural-vlabeled</a>
 
<b>by </b><i><a class="txt" href="lexbfs.html#E11:110"><i><font color="Green" title="E50">A5</font></i></a>, <a class="txt" href="lexbfs.html#E26:110"><i><font color="Green" title="E56">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E39:110"/>
<font color="Maroon" title="c14">F</font> is <a href="lexbfs.html#V11" title="LEXBFS:attr.11">natural-vlabeled</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D17" target="_self" title="LEXBFS:def.17">Def17</a></i>;<br/>

<a NAME="E40:110"/>
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="lexbfs.html#E19:110"><i><font color="Green" title="E53">A8</font></i></a>, <a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></i>;<br/>
<b>hence </b><a NAME="E41:110"/>
<font color="Maroon" title="c14">F</font> is <a href="lexbfs.html#V18" title="LEXBFS:attr.18">vlabel-initially-empty</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D24" target="_self" title="LEXBFS:def.24">Def24</a></i>;<br/>

<a NAME="E42:110"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">F</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  holds <br/> ex <font color="Olive" title="b2">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( not <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> &amp;  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Olive" title="b1">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" title="c15">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:110_7"/><i><font color="Green" title="E62">A24</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">F</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 ;<br/>

<a NAME="E2:110_7"/>
<font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E28:110"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="lexbfs.html#E29:110"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="lexbfs.html#E32:110"><i><font color="Green" title="E61">A22</font></i></a>, <a class="txt" href="lexbfs.html#E1:110_7"><i><font color="Green" title="E62">A24</font></i></a>, <a class="ref" href="glib_000.html#D57" title="GLIB_000:def.57">GLIB_000:def 57</a></i>;<br/>
<a NAME="E3:110_7"/><b>then </b>
( <font color="Maroon" title="c15">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:110_7"/><b>then </b><i><font color="Green" title="E63">A25</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<b>take </b>
<font color="Maroon" title="c1">v</font>
;<br/>

<a NAME="E5:110_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="lexbfs.html#E28:110"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="lexbfs.html#E29:110"><i><font color="Green" title="E59">A16</font></i></a>, <a class="txt" href="lexbfs.html#E32:110"><i><font color="Green" title="E61">A22</font></i></a>, <a class="ref" href="glib_000.html#D57" title="GLIB_000:def.57">GLIB_000:def 57</a></i>;<br/>
<a NAME="E6:110_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="lexbfs.html#E4:110_7"><i><font color="Green" title="E63">A25</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:110_7"/>
( not <font color="Maroon" title="c1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> &amp;  <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="lexbfs.html#K8" title="LEXBFS:func.8">.</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">v</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">F</font> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c15">n</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="lexbfs.html#E7:110"><i><font color="Green" title="E48">A3</font></i></a>, <a class="txt" href="lexbfs.html#E19:110"><i><font color="Green" title="E53">A8</font></i></a>, <a class="txt" href="lexbfs.html#E27:110"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="lexbfs.html#E4:110_7"><i><font color="Green" title="E63">A25</font></i></a>, <a class="ref" href="funct_4.html#T21" title="FUNCT_4:th.21">FUNCT_4:21</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E43:110"/>
<font color="Maroon" title="c14">F</font> is <a href="lexbfs.html#V19" title="LEXBFS:attr.19">adds-one-at-a-step</a>
 <b>by </b><i><a class="ref" href="lexbfs.html#D25" target="_self" title="LEXBFS:def.25">Def25</a></i>;<br/>


</div>
