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

<b>set </b><font color="Maroon">G1</font> = <font color="Maroon">G</font> <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">g</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a> ;<br/>
<a NAME="E1:130_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4">VLabelSelector</a> ,<a href="xboole_0.html#K1">{}</a>  <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="glib_003.html#T8">GLIB_003:8</a></i>;<br/>
<a NAME="E2:130_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E3:130_1_1"/><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="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a> </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">G</font> <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>
 
<b>by </b><i><a class="ref" href="lexbfs.html#T4" target="_self">Th4</a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a>  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"><font color="Maroon">G</font> <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="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T24">PARTFUN1:24</a></i>;<br/>
<b>set </b><font color="Maroon">G2</font> = <span class="p1">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font>;<br/>
<a NAME="E5:130_1_1"/><i><font color="Green">E36</font></i>: 
 <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/>
<a NAME="E6:130_1_1"/><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"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <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>
 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>;<br/>
<a NAME="E7:130_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="glib_003.html#K7">the_VLabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span> is <a href="seqm_3.html#V6">natural-yielding</a>
 
<b>by </b><i><a class="ref" href="lexbfs.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/>
<a NAME="E8:130_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T14">FUNCOP_1:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:130_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:130_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:130_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E4:130_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E10:130_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span></span>)</span> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:130_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E12:130_1_1"/><b>then </b>
(  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span> is <a href="finset_1.html#V2">finite-yielding</a> &amp;  <a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<font color="Maroon">f</font></span>)</span> is <a href="lexbfs.html#V2" target="_self">natsubset-yielding</a> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:130_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">G</font> <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="glib_000.html#K13">.set</a> <a href="lexbfs.html#K5" target="_self">V2LabelSelector</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> is  <a href="lexbfs.html#V3" target="_self">natural-vlabeled</a> <a href="lexbfs.html#V6" target="_self">finite-v2labeled</a> <a href="lexbfs.html#V7" target="_self">natsubset-v2labeled</a> <a href="lexbfs.html#NM3">VVGraph</a>
 <b>by </b><i>, , , </i>;<br/>


</div>
