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

<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">G1</font><br/>  for <font color="Olive" title="b4">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st  not <font color="Olive" title="b3">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v2</font> holds <br/>( ( <font color="Olive" title="b3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Olive" title="b4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Olive" title="b4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Olive" title="b3">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Olive" title="b3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Olive" title="b4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Olive" title="b4">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Olive" title="b3">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">v1</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G1</font><br/>  for <font color="Olive" title="b3">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  st  not <font color="Olive" title="b2">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> holds <br/>( ( <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Olive" title="b3">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Olive" title="b2">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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="c1">G2</font>  st  not <font color="Olive" title="b1">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</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="b2">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Olive" title="b1">v1</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>  ) &amp; ( <font color="Olive" title="b2">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Olive" title="b1">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c1">G2</font>  st  not <font color="Maroon" title="c4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> holds <br/>( ( <font color="Maroon" title="c4">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="Olive" title="b1">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">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>  ) &amp; ( <font color="Olive" title="b1">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> implies ( ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) ) )</span><br/>









<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
(  not <font color="Maroon" title="c4">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1"/>
( <font color="Maroon" title="c2">V</font> <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">G2</font> or  not <font color="Maroon" title="c2">V</font> <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">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1"/>
<font color="Maroon" title="c2">V</font> <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">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon" title="c7">f</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:61_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c6">E</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">E</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">G1</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">G2</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c7">f</font> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">G1</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">G2</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c7">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E4:61_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c1">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_1_1"/>
( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="glib_000.html#T83" onmouseover="rs('glib_000/T83')" onmouseout="rh()">GLIB_000:83</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E6:61_1_1"/>
( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  iff <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_1_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:61_1_1_1"/>
 not <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:61_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T113" onmouseover="rs('glib_000/T113')" onmouseout="rh()">GLIB_000:113</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:61_1_1_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:61_1_1_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="glib_000.html#T71" onmouseover="rs('glib_000/T71')" onmouseout="rh()">GLIB_000:71</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1_1_1"/>
( <font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c1">G2</font> or  not <font color="Maroon" title="c9">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> <font color="Maroon" title="c1">G2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E7:61_1_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:61_1_1_1_1_1"/>
<font color="Maroon" title="c8">w</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E3:61_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="glib_000.html#T71" onmouseover="rs('glib_000/T71')" onmouseout="rh()">GLIB_000:71</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="glib_000.html#T113" onmouseover="rs('glib_000/T113')" onmouseout="rh()">GLIB_000:113</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="61_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 not <font color="Maroon" title="c9">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> <font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:61_1_1_1_1_2"/>
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c6">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E7:61_1_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:61_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:61_1_1_1_1_2"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="xboole_0.html#T5" onmouseover="rs('xboole_0/T5')" onmouseout="rh()">XBOOLE_0:5</a></span>;<br/><a NAME="E4:61_1_1_1_1_2"/><span class="kw">then </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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> &amp; <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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/><a NAME="E5:61_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E6:61_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E3:61_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">thus </span><a NAME="E7:61_1_1"/>
( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_1_2"/><span class="lab"><font color="Green" title="E32">A10</font></span>: 
<font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> </span><br/>

<span class="kw">assume </span><a NAME="E2:61_1_1_2"/>
 not <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:61_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A11</font></span>: 
<font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:61_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E1:61_1_1_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="glib_000.html#T84" onmouseover="rs('glib_000/T84')" onmouseout="rh()">GLIB_000:84</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:61_1_1_2"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
( <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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="c8">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c3">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61_1_1_2"><span class="lab"><font color="Green" title="E32">A10</font></span></a>, <a class="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>;<br/>
<a NAME="E6:61_1_1_2"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:61_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
( <font color="Maroon" title="c8">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> <font color="Maroon" title="c3">G1</font> &amp; ( <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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T61" onmouseover="rs('glib_000/T61')" onmouseout="rh()">GLIB_000:61</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1_2_1"/><span class="kw">then </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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A14</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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">w</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font>;<br/><a NAME="E2:61_1_1_2_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E1:61_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:61_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v2</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E5:61_1_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="glib_012.html#T17" target="_self" onmouseover="rs('glib_012/T17')" onmouseout="rh()">Th17</a></span>;<br/><a NAME="E4:61_1_1_2_1_1"/><span class="kw">then </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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T71" onmouseover="rs('glib_000/T71')" onmouseout="rh()">GLIB_000:71</a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="glib_000.html#T113" onmouseover="rs('glib_000/T113')" onmouseout="rh()">GLIB_000:113</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="61_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_2_1_2"/><span class="lab"><font color="Green" title="E36">A15</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="c3">G1</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">w</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font>;<br/><a NAME="E2:61_1_1_2_1_2"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:61_1_1_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E1:61_1_1_2_1_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:61_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v2</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font>,<font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E5:61_1_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="glib_012.html#T17" target="_self" onmouseover="rs('glib_012/T17')" onmouseout="rh()">Th17</a></span>;<br/><a NAME="E4:61_1_1_2_1_2"/><span class="kw">then </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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K38" title="GLIB_000:func.38">.allNeighbors()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T71" onmouseover="rs('glib_000/T71')" onmouseout="rh()">GLIB_000:71</a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_2"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="glib_000.html#T113" onmouseover="rs('glib_000/T113')" onmouseout="rh()">GLIB_000:113</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">thus </span><a NAME="E8:61_1_1"/>
( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_1_3"/>
<font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:61_1_1_3"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
( <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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="c8">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c1">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  implies <font color="Olive" title="b1">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> ) &amp; ( <font color="Olive" title="b1">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> implies <font color="Olive" title="b1">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">e0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  implies <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> ) &amp; ( <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> implies <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  ) )</span><br/><div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> implies <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</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="E1:61_1_1_3_1_1"/>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font></span><br/><a NAME="E2:61_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A17</font></span>: 
( <font color="Maroon" title="c9">e0</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> <font color="Maroon" title="c3">G1</font> &amp; ( <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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T61" onmouseover="rs('glib_000/T61')" onmouseout="rh()">GLIB_000:61</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:61_1_1_3_1_1_1"/><span class="kw">then </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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A18</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">w</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font>;<br/><a NAME="E2:61_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c9">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E1:61_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:61_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c9">e0</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> <font color="Maroon" title="c1">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_1_3_1_1_1_1_1"/>
 not <font color="Maroon" title="c9">e0</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> <font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:61_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:61_1_1_3_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:61_1_1_3_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:61_1_1_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="61_1_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A21</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">w</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font>;<br/><a NAME="E2:61_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A22</font></span>: 
<font color="Maroon" title="c9">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</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="c3">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font>,<font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:61_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E1:61_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E34">A21</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E3:61_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c9">e0</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> <font color="Maroon" title="c1">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="61_1_1_3_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:61_1_1_3_1_1_1_2_1"/>
 not <font color="Maroon" title="c9">e0</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> <font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:61_1_1_3_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A23</font></span>: 
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E3:61_1_1_3_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:61_1_1_3_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:61_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E36">A23</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:61_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:61_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E35">A22</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a>, <a class="ref" href="glib_000.html#T62" onmouseover="rs('glib_000/T62')" onmouseout="rh()">GLIB_000:62</a></span>;<br/><span class="kw">hence </span><a NAME="E5:61_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">assume </span><a NAME="E2:61_1_1_3_1"/>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span><br/><a NAME="E3:61_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:61_1_1_3_1"/>
<font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:61_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="glib_000.html#T78" onmouseover="rs('glib_000/T78')" onmouseout="rh()">GLIB_000:78</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E5:61_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A24</font></span>: 
<font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.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="c8">e</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:61_1_1_3"/>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:61_1_1_3"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c3">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:61_1_1_3"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:61_1_1_3"/>
<font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:61_1_1_3"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="ref" href="glib_000.html#D51" onmouseover="rs('glib_000/D51')" onmouseout="rh()">GLIB_000:def 51</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="61_1_2"><span class="kw">suppose </span></a><a NAME="E1:61_1_2"/>
 not <font color="Maroon" title="c2">V</font> <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">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><div class="add"><a NAME="E2:61_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:61_1_2"/>
( ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) &amp; ( <font color="Maroon" title="c5">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  implies <font color="Maroon" title="c4">v1</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="glib_000.html#T97" onmouseover="rs('glib_000/T97')" onmouseout="rh()">GLIB_000:97</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
