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<div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:56_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#K21" title="GLIB_000:func.21">.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#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c2">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
( <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#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c2">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G2</font> being  <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a>   <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">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" title="b1">G2</font> is  <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a>  &amp; <font color="Olive" title="b1">G2</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a>  )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c4">X</font> being   <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a>   <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">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/><span class="kw">set </span><font color="Maroon" title="c5">W</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_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_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span>;<br/><a NAME="E3:56_1_1_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E4:56_1_1_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_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#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">W</font> = <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_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_Edges_of</a> <font color="Maroon" title="c4">X</font></span>)</span> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c7">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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">VL9</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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p4">(<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p4">(<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">VL9</font> = <font color="Maroon" title="c9">VL9</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p4">(<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a>, <a class="ref" href="relset_1.html#T13" title="RELSET_1:th.13">RELSET_1:13</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</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="c10">VL9</font>;<br/><a NAME="E8:56_1_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</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="c10">VL9</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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_003.html#E14"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/><a NAME="E9:56_1_1_1_1_1"/>
<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_003.html#E14"><span class="lab"><font color="Green" title="E7">Lm3</font></span></a></span>;<br/><a NAME="E10:56_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</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="c10">VL9</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="glib_000.html#T88" title="GLIB_000:th.88">GLIB_000:88</a></span>;<br/><a NAME="E11:56_1_1_1_1_1"/><span class="kw">then </span>
<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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</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="c10">VL9</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T90" title="GLIB_000:th.90">GLIB_000:90</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</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="c10">VL9</font> as    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T46" title="GLIB_000:th.46">GLIB_000:46</a></span>;<br/><span class="lab"><font color="Green" title="E14">A4</font></span>:  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c12">G2</font> = 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></span>
<br/>.= 
<font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>
;<br/> <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c12">G2</font> = 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c4">X</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></span>
<br/>.= 
<font color="Maroon" title="c3">E</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">G2</font> = <font color="Maroon" title="c12">G2</font> as   <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a>   <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E13:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">G2</font> = <font color="Maroon" title="c13">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">G2</font> is  <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a>  &amp; <font color="Maroon" title="c14">G2</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a>  )</span><br/> <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c14">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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="glib_003.html#K2" title="GLIB_003:func.2">WeightSelector</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T12" title="GLIB_000:th.12">GLIB_000:12</a></span>
<br/>.= 
<font color="Maroon" title="c6">W</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_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_Edges_of</a> <font color="Maroon" title="c14">G2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></span>
;<br/><span class="kw">hence </span><a NAME="E17:56_1_1_1_1_1"/>
<font color="Maroon" title="c14">G2</font> is  <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_003.html#D10" target="_self" title="GLIB_003:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">G2</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a> </span><br/> <a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <font color="Maroon" title="c14">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#K2" title="GLIB_003:func.2">WeightSelector</a> ,<font color="Maroon" title="c6">W</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T11" title="GLIB_000:th.11">GLIB_000:11</a></span>
<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#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c14">G2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></span>
;<br/><span class="kw">hence </span><a NAME="E19:56_1_1_1_1_1"/>
<font color="Maroon" title="c14">G2</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_003.html#D12" target="_self" title="GLIB_003:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:56_1_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
(  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#K21" title="GLIB_000:func.21">.edgesBetween</a> <font color="Maroon" title="c2">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">GG</font> being  <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a>   <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">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" title="b1">GG</font> is  <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a>  &amp; <font color="Olive" title="b1">GG</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c4">GG</font> = <font color="Maroon" title="c1">G</font> as    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T43" title="GLIB_000:th.43">GLIB_000:43</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">GG</font> = <font color="Maroon" title="c4">GG</font> as   <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a>   <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">inducedSubgraph</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="ref" href="glib_000.html#D39" title="GLIB_000:def.39">GLIB_000:def 39</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c6">GG</font> = <font color="Maroon" title="c5">GG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">GG</font> is  <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a>  &amp; <font color="Maroon" title="c6">GG</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a>  )</span><br/><a NAME="E4:56_1_1_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E5:56_1_1_1_1_2"/><span class="kw">then </span>
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_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#K5" title="GLIB_003:func.5">the_Weight_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></span>;<br/><span class="kw">hence </span><a NAME="E6:56_1_1_1_1_2"/>
<font color="Maroon" title="c6">GG</font> is  <a href="glib_003.html#V4" title="GLIB_003:attr.4">weight-inheriting</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_003.html#D10" target="_self" title="GLIB_003:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">GG</font> is  <a href="glib_003.html#V6" title="GLIB_003:attr.6">vlabel-inheriting</a> </span><br/><a NAME="E7:56_1_1_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_003.html#E13"><span class="lab"><font color="Green" title="E6">Lm2</font></span></a></span>;<br/><a NAME="E8:56_1_1_1_1_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>;<br/><span class="kw">hence </span><a NAME="E9:56_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_003.html#D12" target="_self" title="GLIB_003:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:56_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_003.html#V1" title="GLIB_003:attr.1">[Weighted]</a>   <a href="glib_003.html#V3" title="GLIB_003:attr.3">[VLabeled]</a>   <a href="glib_000.html#M2" title="GLIB_000:mode.2">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#V4" title="GLIB_003:attr.4">weight-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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
