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<b>set </b><font color="Maroon" title="c2">Z</font> = <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c3">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>;<br/>
<a NAME="E1:21_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> holds <br/><font color="Olive" title="b1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">Y</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T25" title="FINSET_1:th.25">FINSET_1:25</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">X</font> =  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ,<font color="Maroon" title="c4">Y</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ,<font color="Maroon" title="c4">Y</font> )</span><br/><b>assume </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ,<font color="Maroon" title="c4">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c7">G2</font> being   <a href="glib_003.html#NM8" title="GLIB_003:NM.8">WSubgraph</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:21_1_1_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G2</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  )
 <b>by </b><i><a class="ref" href="glib_004.html#D5" target="_self" title="GLIB_004:def.5">Def10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G2</font> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> )</span><br/><b>assume </b><a NAME="E1:21_1_1_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:21_1_1_1_1"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a>  &amp; <font color="Maroon" title="c7">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_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="E1:21_1_1_1_1_1_1"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K2" title="GLIB_003:func.2">WeightSelector</a>  )
 </span><b>by </b><i><a class="txt" href="#E3:21_1_1_1_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c7">G2</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/><a NAME="E3:21_1_1_1_1_1_1_1"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E2:21_1_1_1_1_1_1_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c7">G2</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/><a NAME="E3:21_1_1_1_1_1_1_2"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>hence </b><a NAME="E4:21_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E2:21_1_1_1_1_1_1_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">G2</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/><a NAME="E3:21_1_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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="c7">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T48" title="GLIB_000:th.48">GLIB_000:48</a></i>;<br/><a NAME="E4:21_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/><a NAME="E5:21_1_1_1_1_1_1_3"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E4:21_1_1_1_1_1_1_3"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_1_4"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c7">G2</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/><a NAME="E3:21_1_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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="c7">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T48" title="GLIB_000:th.48">GLIB_000:48</a></i>;<br/><a NAME="E4:21_1_1_1_1_1_1_4"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/><a NAME="E5:21_1_1_1_1_1_1_4"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_1_1_1_4"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E4:21_1_1_1_1_1_1_4"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:21_1_1_1_1_1_1_5"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K2" title="GLIB_003:func.2">WeightSelector</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font></span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c7">G2</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1_1"><i><font color="Green" title="E17">A3</font></i></a></i>;<br/><a NAME="E3:21_1_1_1_1_1_1_5"/><b>then </b>
<font color="Maroon" title="c8">y</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#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c7">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#D10" title="GLIB_003:def.10">GLIB_003:def 10</a></i>;<br/><a NAME="E4:21_1_1_1_1_1_1_5"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/><a NAME="E5:21_1_1_1_1_1_1_5"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</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#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p3">(<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></span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#D3" title="ENUMSET1:def.3">ENUMSET1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1_1_1_1_5"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E4:21_1_1_1_1_1_1_5"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:21_1_1_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:21_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E6:21_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ,<font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="txt" href="#E3:21_1_1_1"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:21_1_1"/><b>then </b>
<font color="Maroon" title="c1">G</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <a href="glib_004.html#K2" title="GLIB_004:func.2">WGraphSelectors</a> ,<font color="Maroon" title="c4">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:21_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_004.html#K3" title="GLIB_004:func.3">.allWSubgraphs()</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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