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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<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="E1:191"/><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="E2:191"/><i><font color="Green">E36</font></i>: 
 <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="E3:191"/><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="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/>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="191_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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="191_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:191_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="191_1_1_1"><b>suppose </b></a><a NAME="E1:191_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:191_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="191_1_1_2"><b>suppose </b></a><a NAME="E1:191_1_1_2"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:191_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>thus </b><a NAME="E6:191"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K20" target="_self">MCS:Init</a> <font color="Maroon">G</font></span>)</span></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="glib_000.html#T11">GLIB_000:11</a>, </i>;<br/>

<a NAME="E7:191"/>
 <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/>
<b>hence </b><a NAME="E8:191"/>
<span class="p1">(<span class="default"><a href="lexbfs.html#K6" target="_self">the_V2Label_of</a> <span class="p2">(<span class="default"><a href="lexbfs.html#K20" target="_self">MCS:Init</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


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