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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1_1_1_1"/>
( ( <font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">V</font> ) or  not <font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> or  not <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> &amp; <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">V</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">X</font> being  <a href="glib_003.html#V1" target="_self">[Weighted]</a> <a href="glib_003.html#V2" target="_self">[ELabeled]</a> <a href="glib_003.html#V3" target="_self">[VLabeled]</a>  <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font>,<font color="Maroon">E</font>;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font></span>)</span>;<br/><a NAME="E3:59_1_1_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:59_1_1_1_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_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>then reconsider </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>set </b><font color="Maroon">G1</font> = <font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font>;<br/><a NAME="E6:59_1_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">EL</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">EL'</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span></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_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T24">PARTFUN1:24</a></i>;<br/><a NAME="E8:59_1_1_1_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_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/><b>then reconsider </b><font color="Maroon">EL'</font> = <font color="Maroon">EL'</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></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_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T28">PARTFUN1:28</a></i>;<br/><b>set </b><font color="Maroon">G2</font> = <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font>;<br/><a NAME="E10:59_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font>
 <b>by </b><i/>;<br/><a NAME="E11:59_1_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><b>set </b><font color="Maroon">VL</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">VL'</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span></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_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T24">PARTFUN1:24</a></i>;<br/><a NAME="E13:59_1_1_1_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_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/><b>then reconsider </b><font color="Maroon">VL'</font> = <font color="Maroon">VL'</font> 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"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></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_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T28">PARTFUN1:28</a></i>;<br/><b>set </b><font color="Maroon">G3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL'</font>;<br/><a NAME="E15:59_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL'</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font>
 <b>by </b><i/>;<br/><a NAME="E16:59_1_1_1_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL'</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E17:59_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL'</font> is    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T90">GLIB_000:90</a></i>;<br/><b>then reconsider </b><font color="Maroon">G3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL'</font> as    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T46">GLIB_000:46</a></i>;<br/><i><font color="Green">E57</font></i>:  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font> = 
 <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
<font color="Maroon">V</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>
;<br/> <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G3</font> = 
 <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
<br/>.= 
<font color="Maroon">E</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>
;<br/><b>then reconsider </b><font color="Maroon">G3</font> = <font color="Maroon">G3</font> as  <a href="glib_003.html#V1" target="_self">[Weighted]</a> <a href="glib_003.html#V2" target="_self">[ELabeled]</a> <a href="glib_003.html#V3" target="_self">[VLabeled]</a>  <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font>,<font color="Maroon">E</font> <b>by </b><i>, , <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>take </b><font color="Maroon">G3</font> = <font color="Maroon">G3</font>;<br/> <a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G3</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
<font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/><b>hence </b><a NAME="E23:59_1_1_1_1_1"/>
<font color="Maroon">G3</font> is <a href="glib_003.html#V4" target="_self">weight-inheriting</a>
 <b>by </b><i/>;<br/><a NAME="E24:59_1_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G3</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/> <a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G3</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/><b>hence </b><a NAME="E26:59_1_1_1_1_1"/>
<font color="Maroon">G3</font> is <a href="glib_003.html#V5" target="_self">elabel-inheriting</a>
 <b>by </b><i/>;<br/> <a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G3</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G3</font></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>
;<br/><b>hence </b><a NAME="E28:59_1_1_1_1_1"/>
<font color="Maroon">G3</font> is <a href="glib_003.html#V6" target="_self">vlabel-inheriting</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:59_1_1_1_1_2"/><i><font color="Green">E60</font></i>: 
(  not <font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> or  not <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> <a href="glib_000.html#K22">.edgesBetween</a> <font color="Maroon">V</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">GG</font> = <font color="Maroon">G</font> as    <a href="glib_000.html#M2">Subgraph</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T43">GLIB_000:43</a></i>;<br/><b>reconsider </b><font color="Maroon">GG</font> = <font color="Maroon">GG</font> as  <a href="glib_003.html#V1" target="_self">[Weighted]</a> <a href="glib_003.html#V2" target="_self">[ELabeled]</a> <a href="glib_003.html#V3" target="_self">[VLabeled]</a>  <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font>,<font color="Maroon">E</font> <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>take </b><font color="Maroon">GG</font> = <font color="Maroon">GG</font>;<br/><a NAME="E4:59_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:59_1_1_1_1_2"/><b>then </b>
 <a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>;<br/><b>hence </b><a NAME="E6:59_1_1_1_1_2"/>
<font color="Maroon">GG</font> is <a href="glib_003.html#V4" target="_self">weight-inheriting</a>
 <b>by </b><i/>;<br/><a NAME="E7:59_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E8:59_1_1_1_1_2"/><b>then </b>
 <a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><b>hence </b><a NAME="E9:59_1_1_1_1_2"/>
<font color="Maroon">GG</font> is <a href="glib_003.html#V5" target="_self">elabel-inheriting</a>
 <b>by </b><i/>;<br/><a NAME="E10:59_1_1_1_1_2"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E11:59_1_1_1_1_2"/><b>then </b>
 <a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" target="_self">the_VLabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><b>hence </b><a NAME="E12:59_1_1_1_1_2"/>
<font color="Maroon">GG</font> is <a href="glib_003.html#V6" target="_self">vlabel-inheriting</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="glib_003.html#V1" target="_self">[Weighted]</a> <a href="glib_003.html#V2" target="_self">[ELabeled]</a> <a href="glib_003.html#V3" target="_self">[VLabeled]</a>  <a href="glib_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font>,<font color="Maroon">E</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V4" target="_self">weight-inheriting</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V5" target="_self">elabel-inheriting</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V6" target="_self">vlabel-inheriting</a> )
 ;<br/>


</div>
