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<div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_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="E1:58_1_1_1_1"/>
( ( <font color="Maroon" title="c2">V</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c2">V</font> ) or  not <font color="Maroon" title="c2">V</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> or  not <font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c2">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:58_1_1_1_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c2">V</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> &amp; <font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c2">V</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">X</font> being  <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>  <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">E</font>;<br/><b>set </b><font color="Maroon" title="c5">EL</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">EL'</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <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="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T24" title="PARTFUN1:th.24">PARTFUN1:24</a></i>;<br/><a NAME="E4:58_1_1_1_1_1"/>
 <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="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">EL'</font> = <font color="Maroon" title="c6">EL'</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T28" title="PARTFUN1:th.28">PARTFUN1:28</a></i>;<br/><b>set </b><font color="Maroon" title="c8">G1</font> = <font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font>;<br/><a NAME="E6:58_1_1_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E14"><i><font color="Green" title="E7">Lm3</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c9">VL</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c10">VL'</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <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="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span></span>)</span>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T24" title="PARTFUN1:th.24">PARTFUN1:24</a></i>;<br/><a NAME="E8:58_1_1_1_1_1"/>
 <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="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span></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"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">VL'</font> = <font color="Maroon" title="c10">VL'</font> 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"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span>, <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="partfun1.html#T28" title="PARTFUN1:th.28">PARTFUN1:28</a></i>;<br/><b>set </b><font color="Maroon" title="c12">G2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</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="c11">VL'</font>;<br/><a NAME="E10:58_1_1_1_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</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="c11">VL'</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E14"><i><font color="Green" title="E7">Lm3</font></i></a></i>;<br/><a NAME="E11:58_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</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="c11">VL'</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E6:58_1_1_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="glib_000.html#T88" title="GLIB_000:th.88">GLIB_000:88</a></i>;<br/><a NAME="E12:58_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</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="c11">VL'</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T90" title="GLIB_000:th.90">GLIB_000:90</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">G2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</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="c11">VL'</font> as    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T46" title="GLIB_000:th.46">GLIB_000:46</a></i>;<br/><i><font color="Green" title="E15">A5</font></i>:  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c13">G2</font> = 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="glib_003.html#E11:58_1_1_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
<br/>.= 
<font color="Maroon" title="c2">V</font>
<b>by </b><i><a class="txt" href="glib_003.html#E1:58_1_1_1_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>
;<br/> <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c13">G2</font> = 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c4">X</font>
<b>by </b><i><a class="txt" href="glib_003.html#E11:58_1_1_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
<br/>.= 
<font color="Maroon" title="c3">E</font>
<b>by </b><i><a class="txt" href="glib_003.html#E1:58_1_1_1_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c14">G2</font> = <font color="Maroon" title="c13">G2</font> as  <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>  <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">E</font> <b>by </b><i><a class="txt" href="glib_003.html#E1:58_1_1_1_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="glib_003.html#E14:58_1_1_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>take </b><font color="Maroon" title="c15">G2</font> = <font color="Maroon" title="c14">G2</font>;<br/> <a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c15">G2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12" title="GLIB_000:th.12">GLIB_000:12</a></i>
<br/>.= 
<font color="Maroon" title="c7">EL'</font>
<b>by </b><i><a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c15">G2</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_003.html#E11:58_1_1_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
;<br/><b>hence </b><a NAME="E18:58_1_1_1_1_1"/>
<font color="Maroon" title="c15">G2</font> is <a href="glib_003.html#V5" title="GLIB_003:attr.5">elabel-inheriting</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D11" target="_self" title="GLIB_003:def.11">Def11</a></i>;<br/> <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c15">G2</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">X</font> <a href="glib_000.html#K13" title="GLIB_000:func.13">.set</a> <a href="glib_003.html#K3" title="GLIB_003:func.3">ELabelSelector</a> ,<font color="Maroon" title="c7">EL'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c15">G2</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_003.html#E10:58_1_1_1_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
;<br/><b>hence </b><a NAME="E20:58_1_1_1_1_1"/>
<font color="Maroon" title="c15">G2</font> is <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D12" target="_self" title="GLIB_003:def.12">Def12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:58_1_1_1_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
(  not <font color="Maroon" title="c2">V</font> is  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> or  not <font color="Maroon" title="c3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <font color="Maroon" title="c2">V</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c4">GG</font> = <font color="Maroon" title="c1">G</font> as    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="glib_000.html#T43" title="GLIB_000:th.43">GLIB_000:43</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">GG</font> = <font color="Maroon" title="c4">GG</font> as  <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>  <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">E</font> <b>by </b><i><a class="txt" href="glib_003.html#E1:58_1_1_1_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></i>;<br/><b>take </b><font color="Maroon" title="c6">GG</font> = <font color="Maroon" title="c5">GG</font>;<br/><a NAME="E4:58_1_1_1_1_2"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E12"><i><font color="Green" title="E5">Lm1</font></i></a></i>;<br/><a NAME="E5:58_1_1_1_1_2"/><b>then </b>
 <a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6" title="GLIB_003:func.6">the_ELabel_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/><b>hence </b><a NAME="E6:58_1_1_1_1_2"/>
<font color="Maroon" title="c6">GG</font> is <a href="glib_003.html#V5" title="GLIB_003:attr.5">elabel-inheriting</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D11" target="_self" title="GLIB_003:def.11">Def11</a></i>;<br/><a NAME="E7:58_1_1_1_1_2"/>
 <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> <font color="Maroon" title="c1">G</font></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> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_003.html#E13"><i><font color="Green" title="E6">Lm2</font></i></a></i>;<br/><a NAME="E8:58_1_1_1_1_2"/><b>then </b>
 <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c1">G</font> <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> <font color="Maroon" title="c1">G</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/><b>hence </b><a NAME="E9:58_1_1_1_1_2"/>
<font color="Maroon" title="c6">GG</font> is <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a>
 <b>by </b><i><a class="ref" href="glib_003.html#D12" target="_self" title="GLIB_003:def.12">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:58_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being <a href="glib_003.html#V2" title="GLIB_003:attr.2">[ELabeled]</a> <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>  <a href="glib_000.html#M3" title="GLIB_000:mode.3">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">E</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V5" title="GLIB_003:attr.5">elabel-inheriting</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a> )
 ;<br/>


</div>
