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

<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> holds <br/>( ( <font color="Olive" title="b1">v1</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> implies <font color="Olive" title="b2">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> ) &amp; ( <font color="Olive" title="b1">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> implies <font color="Olive" title="b2">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> holds <br/>( ( <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> implies <font color="Olive" title="b1">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> ) &amp; ( <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> implies <font color="Olive" title="b1">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> ) )</span><br/><b>let </b><font color="Maroon" title="c4">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">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="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> implies ( ( <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> implies <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> ) &amp; ( <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> implies <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> ) ) )</span><br/>







<b>assume </b><a NAME="E1:189"/><i><font color="Green" title="E77">A1</font></i>: 
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">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="c3">v1</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> implies <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> ) &amp; ( <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> implies <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> implies <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:189_1"/>
<font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a></span><br/><a NAME="E2:189_1"/><b>then </b>
<font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D51" target="_self" title="GLIB_000:def.51">Def51</a></i>;<br/><a NAME="E3:189_1"/><b>then </b>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="glib_000.html#E1:189"><i><font color="Green" title="E77">A1</font></i></a>, <a class="ref" href="glib_000.html#T99" target="_self" title="GLIB_000:th.99">Th99</a></i>;<br/><b>hence </b><a NAME="E4:189_1"/>
<font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D51" target="_self" title="GLIB_000:def.51">Def51</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>

<b>assume </b><a NAME="E3:189"/>
<font color="Maroon" title="c3">v1</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a></span><br/>

<b>then consider </b><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:189"/><i><font color="Green" title="E78">A2</font></i>: 
( <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D53" target="_self" title="GLIB_000:def.53">Def53</a></i>;<br/>
<a NAME="E6:189"/>
( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c4">v2</font>,<font color="Maroon" title="c2">G2</font> )
 
<b>by </b><i><a class="txt" href="glib_000.html#E1:189"><i><font color="Green" title="E77">A1</font></i></a>, <a class="txt" href="glib_000.html#E5:189"><i><font color="Green" title="E78">A2</font></i></a>, <a class="ref" href="glib_000.html#T91" target="_self" title="GLIB_000:th.91">Th91</a>, <a class="ref" href="glib_000.html#T99" target="_self" title="GLIB_000:th.99">Th99</a></i>;<br/>
<b>hence </b><a NAME="E7:189"/>
<font color="Maroon" title="c4">v2</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 <b>by </b><i><a class="ref" href="glib_000.html#D53" target="_self" title="GLIB_000:def.53">Def53</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>
