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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_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="E1:52_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="52_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:52_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_000.html#M3">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font>,<font color="Maroon">E</font>;<br/><b>consider </b><font color="Maroon">W</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font>;<br/><b>set </b><font color="Maroon">G2</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/><b>consider </b><font color="Maroon">EL</font> being   <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="numbers.html#K1">REAL</a> ;<br/><b>set </b><font color="Maroon">G3</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/><b>consider </b><font color="Maroon">VL</font> being   <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="numbers.html#K1">REAL</a> ;<br/><b>set </b><font color="Maroon">G4</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="E6:52_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> &amp; <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> <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> &amp; <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> <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> )
 <b>by </b><i/>;<br/><a NAME="E7:52_1_1_1_1_1"/><b>then </b>
<font color="Maroon">X</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><a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E8:52_1_1_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">X</font> <a href="glib_000.html#R5">==</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="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E9:52_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">G4</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/><a NAME="E11:52_1_1_1_1_1"/>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><a NAME="E12:52_1_1_1_1_1"/><b>then </b>
(  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">G4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp;  <a href="glib_000.html#K26">the_Edges_of</a> <font color="Maroon">G4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E13:52_1_1_1_1_1"/><b>then </b>
<font color="Maroon">G4</font> is    <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>hence </b><a NAME="E14:52_1_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <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#V1" target="_self">[Weighted]</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V2" target="_self">[ELabeled]</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V3" target="_self">[VLabeled]</a> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:52_1_1_1_1_2"/><i><font color="Green">E52</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>consider </b><font color="Maroon">W</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>;<br/><b>set </b><font color="Maroon">G2</font> = <font color="Maroon">G</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/><b>consider </b><font color="Maroon">EL</font> being   <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">G</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="numbers.html#K1">REAL</a> ;<br/><b>set </b><font color="Maroon">G3</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#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/><b>consider </b><font color="Maroon">VL</font> being   <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">G</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="numbers.html#K1">REAL</a> ;<br/><b>set </b><font color="Maroon">G4</font> = <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#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="E5:52_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font> &amp; <font color="Maroon">G</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font> <a href="glib_000.html#R5">==</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#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> &amp; <span class="p1">(<span class="default"><font color="Maroon">G</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> <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#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> )
 <b>by </b><i/>;<br/><a NAME="E6:52_1_1_1_1_2"/><b>then </b>
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</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#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><a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><a NAME="E7:52_1_1_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</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#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>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T88">GLIB_000:88</a></i>;<br/><b>then reconsider </b><font color="Maroon">G4</font> = <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#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#T90">GLIB_000:90</a></i>;<br/><a NAME="E9:52_1_1_1_1_2"/>
<font color="Maroon">G4</font> is    <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>hence </b><a NAME="E10:52_1_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <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#V1" target="_self">[Weighted]</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V2" target="_self">[ELabeled]</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V3" target="_self">[VLabeled]</a> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:52_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <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#V1" target="_self">[Weighted]</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V2" target="_self">[ELabeled]</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V3" target="_self">[VLabeled]</a> )
 ;<br/>


</div>
