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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1_1_1"/><i><font color="Green" title="E28">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/><b>then reconsider </b><font color="Maroon" title="c4">V'</font> = <font color="Maroon" title="c2">V</font> as  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> ;<br/><b>reconsider </b><font color="Maroon" title="c5">E'</font> = <font color="Maroon" title="c3">E</font> as   <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#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="txt" href="glib_000.html#E1:54_1_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon" title="c6">S</font> = <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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font>;<br/><b>set </b><font color="Maroon" title="c7">T</font> = <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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font>;<br/><a NAME="E4:54_1_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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#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> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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#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> )
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:54_1_1_1"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E'</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E'</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_1_1_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E'</font>
 ;<br/><a NAME="E2:54_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:54_1_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:54_1_1_1_1"/><b>then </b>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D29" target="_self" title="GLIB_000:def.29">Def29</a></i>;<br/><b>hence </b><a NAME="E4:54_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:54_1_1_1_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c8">S</font> = <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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c4">V'</font> <b>by </b><i><a class="txt" href="glib_000.html#E5:54_1_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_1_1_2"/><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E'</font>
 ;<br/><a NAME="E2:54_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:54_1_1_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:54_1_1_1_2"/><b>then </b>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" target="_self" title="GLIB_000:def.28">Def28</a></i>;<br/><b>hence </b><a NAME="E4:54_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="glib_000.html#E1:54_1_1_1_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">T</font> = <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="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c5">E'</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c4">V'</font> <b>by </b><i><a class="txt" href="glib_000.html#E5:54_1_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/><b>set </b><font color="Maroon" title="c10">IT</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font>;<br/><a NAME="E10:54_1_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E</font> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c8">S</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c9">T</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T91" title="FINSEQ_4:th.91">FINSEQ_4:91</a></i>;<br/><a NAME="E11:54_1_1_1"/><b>then </b>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <span class="p2">(<span class="default"><a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">IT</font> =  <a href="glib_000.html#K12" title="GLIB_000:func.12">createGraph</a> <font color="Maroon" title="c4">V'</font>,<font color="Maroon" title="c5">E'</font>,<font color="Maroon" title="c8">S</font>,<font color="Maroon" title="c9">T</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="txt" href="glib_000.html#E10:54_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="glib_000.html#D34" target="_self" title="GLIB_000:def.34">Def34</a></i>;<br/><b>take </b><font color="Maroon" title="c12">IT</font> = <font color="Maroon" title="c11">IT</font>;<br/><b>thus </b><a NAME="E13:54_1_1_1"/>
(  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c12">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c12">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">E</font> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T91" title="FINSEQ_4:th.91">FINSEQ_4:91</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E2:54_1_1"/>
<font color="Maroon" title="c1">G</font> is    <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="txt" href="glib_000.html#E38"><i><font color="Green" title="E22">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1"/>
( (  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> ) implies  ex <font color="Olive">b<sub>1</sub></font> being   <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> )
 ;<br/>


</div>
