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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b2">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Olive" title="b1">e</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b1">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</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="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font> )</span><br/>





<div><span class="kw">hereby </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="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font> implies <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:163_1"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 ; <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="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font></span><br/><a NAME="E2:163_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A2</font></span>: 
( <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="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> or <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="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" target="_self" title="GLIB_000:th.64">Th64</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="163_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_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="E1:163_1_1_1"/>
( <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="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> or <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="c2">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="163_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:163_1_1_1_1"/><span class="lab"><font color="Green" title="E65">A3</font></span>: 
<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="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</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="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:163_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">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="Maroon" title="c2">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>;<br/><span class="kw">hence </span><a NAME="E3:163_1_1_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E1:163_1_1_1_1"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="ref" href="glib_000.html#D15" target="_self" title="GLIB_000:def.15">Def15</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="163_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:163_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
<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="c2">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">v</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="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font></span><br/><div class="add"><a NAME="E2:163_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">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="Maroon" title="c2">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:163_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="glib_000.html#D43" target="_self" title="GLIB_000:def.43">Def43</a></span>;<br/><span class="kw">hence </span><a NAME="E3:163_1_1_1_2"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:163_1"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="txt" href="#E2:163_1"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E1:163_1_1_1_2"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="glib_000.html#D15" target="_self" title="GLIB_000:def.15">Def15</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="E4:163_1"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font>
 ; <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>

<span class="kw">assume </span><a NAME="E2:163"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.adj</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</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="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:163"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T65" target="_self" title="GLIB_000:th.65">Th65</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>
