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

<span class="kw">let </span><font color="Maroon" title="c1">G1</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">G2</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</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="Maroon" title="c1">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="Olive" title="b1">G2</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/>( ( <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#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <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#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">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">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#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/>( ( <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#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <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#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><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" 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="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#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 holds <br/>( ( <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#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <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#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/><span class="kw">let </span><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" 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="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#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies ( ( <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#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <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#V11" title="GLIB_000:attr.11">endvertex</a>  ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:182"/><span class="lab"><font color="Green" title="E88">A1</font></span>: 
( <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#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <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="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#V10" title="GLIB_000:attr.10">isolated</a>  ) &amp; ( <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#V11" title="GLIB_000:attr.11">endvertex</a>  ) )</span><br/>

<a NAME="E2:182"/><span class="kw">then </span>
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K14" title="GLIB_000:func.14">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T111" target="_self" onmouseover="rs('glib_012/T111')" onmouseout="rh()">Th111</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">u1</font>, <font color="Maroon" title="c6">u2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:182"/><span class="lab"><font color="Green" title="E89">A2</font></span>: 
( <font color="Maroon" title="c5">u1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">u2</font> &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">u1</font>,<font color="Maroon" title="c6">u2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T60" onmouseover="rs('card_2/T60')" onmouseout="rh()">CARD_2:60</a></span>;<br/>
<a NAME="E5:182"/><span class="lab"><font color="Green" title="E90">A3</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">u1</font> = <font color="Maroon" title="c5">u1</font>, <font color="Maroon" title="c8">u2</font> = <font color="Maroon" title="c6">u2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</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="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#V11" title="GLIB_000:attr.11">endvertex</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:182_1"/>
<font color="Maroon" title="c3">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="c4">v2</font> is  <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:182_1"/><span class="lab"><font color="Green" title="E91">A4</font></span>: 
( <font color="Maroon" title="c3">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="c9">e1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c9">e1</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> )
 <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/><a NAME="E4:182_1"/>
<font color="Maroon" title="c9">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">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="#E3:182_1"><span class="lab"><font color="Green" title="E91">A4</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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">v9</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><span class="kw"> such that </span><br/><a NAME="E6:182_1"/><span class="lab"><font color="Green" title="E92">A5</font></span>: 
<font color="Maroon" title="c9">e1</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="c10">v9</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><a NAME="E7:182_1"/><span class="lab"><font color="Green" title="E93">A6</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:182_1"><span class="lab"><font color="Green" title="E91">A4</font></span></a>, <a class="txt" href="#E6:182_1"><span class="lab"><font color="Green" title="E92">A5</font></span></a></span>;<br/><a NAME="E8:182_1"/>
<font color="Maroon" title="c10">v9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
 ;<br/><a NAME="E9:182_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">v9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><a NAME="E10:182_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A7</font></span>: 
( <font color="Maroon" title="c10">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> or <font color="Maroon" title="c10">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">assume </span><a NAME="E11:182_1"/>
 not <font color="Maroon" title="c4">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"> contradiction</span><br/><a NAME="E12:182_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</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#D49" onmouseover="rs('glib_000/D49')" onmouseout="rh()">GLIB_000:def 49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E14:182_1"/><span class="lab"><font color="Green" title="E95">A8</font></span>: 
<font color="Maroon" title="c11">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="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="c12">u</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E16:182_1"/><span class="lab"><font color="Green" title="E96">A9</font></span>: 
<font color="Maroon" title="c11">e2</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="c12">u</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:182_1"><span class="lab"><font color="Green" title="E95">A8</font></span></a>, <a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_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="E1:182_1_1"/>
( ( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> ) or ( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> ) or ( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> ) or ( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:182_1_1_1"/>
( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</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">hence </span><a NAME="E2:182_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:182_1"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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="182_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:182_1_1_2"/>
( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</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">hence </span><a NAME="E2:182_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E6:182_1"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="txt" href="#E7:182_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E10:182_1"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E16:182_1"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="glib_012.html#T100" target="_self" onmouseover="rs('glib_012/T100')" onmouseout="rh()">Th100</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="182_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:182_1_1_3"/>
( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</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">hence </span><a NAME="E2:182_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E6:182_1"><span class="lab"><font color="Green" title="E92">A5</font></span></a>, <a class="txt" href="#E7:182_1"><span class="lab"><font color="Green" title="E93">A6</font></span></a>, <a class="txt" href="#E10:182_1"><span class="lab"><font color="Green" title="E94">A7</font></span></a>, <a class="txt" href="#E16:182_1"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="glib_012.html#T100" target="_self" onmouseover="rs('glib_012/T100')" onmouseout="rh()">Th100</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="182_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:182_1_1_4"/>
( <font color="Maroon" title="c4">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> &amp; <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</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">hence </span><a NAME="E2:182_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:182_1"><span class="lab"><font color="Green" title="E96">A9</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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="E8:182"/><span class="lab"><font color="Green" title="E91">A10</font></span>: 
<font color="Maroon" title="c3">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">v2</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="182_2"><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="E1:182_2"/>
( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font> or <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E5:182"><span class="lab"><font color="Green" title="E90">A3</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_2_1"><span class="kw">suppose </span></a><a NAME="E1:182_2_1"/><span class="lab"><font color="Green" title="E92">A11</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</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">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> </span><br/><div class="add"><a NAME="E2:182_2_1"/>
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</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="c8">u2</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:182"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="ref" href="glib_000.html#T143" onmouseover="rs('glib_000/T143')" onmouseout="rh()">GLIB_000:143</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:182_2_1"/><span class="lab"><font color="Green" title="E93">A12</font></span>: 
<font color="Maroon" title="c9">e2</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="c8">u2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E1:182_2_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><a NAME="E5:182_2_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">e</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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> )</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="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> implies <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</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="E1:182_2_1_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</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="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">v9</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E3:182_2_1_1_1"/><span class="lab"><font color="Green" title="E94">A13</font></span>: 
<font color="Maroon" title="c10">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="c11">v9</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><a NAME="E4:182_2_1_1_1"/>
<font color="Maroon" title="c11">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:182_2_1_1_1_1"/>
<font color="Maroon" title="c11">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">u2</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="E2:182_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:182_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E1:182_2_1"><span class="lab"><font color="Green" title="E92">A11</font></span></a>, <a class="txt" href="#E3:182_2_1_1_1"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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><span class="kw">hence </span><a NAME="E5:182_2_1_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E4:182_2_1"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="txt" href="#E3:182_2_1_1_1"><span class="lab"><font color="Green" title="E94">A13</font></span></a>, <a class="ref" href="glib_000.html#D20" onmouseover="rs('glib_000/D20')" onmouseout="rh()">GLIB_000:def 20</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>

<span class="kw">reconsider </span><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c8">u2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:182_2_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span><br/>

<a NAME="E4:182_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">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="c11">w</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c10">e</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E4:182_2_1"><span class="lab"><font color="Green" title="E93">A12</font></span></a>, <a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:182_2_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</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="E6:182_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="c9">e2</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/><span class="kw">hence </span><a NAME="E7:182_2_1"/>
<font color="Maroon" title="c4">v2</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="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="182_2_2"><span class="kw">suppose </span></a><a NAME="E1:182_2_2"/><span class="lab"><font color="Green" title="E92">A14</font></span>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</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">v2</font> is  <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a> </span><br/><div class="add"><a NAME="E2:182_2_2"/>
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</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="c7">u1</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:182"><span class="lab"><font color="Green" title="E91">A10</font></span></a>, <a class="ref" href="glib_000.html#T143" onmouseover="rs('glib_000/T143')" onmouseout="rh()">GLIB_000:143</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:182_2_2"/><span class="lab"><font color="Green" title="E93">A15</font></span>: 
<font color="Maroon" title="c9">e2</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="c7">u1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="txt" href="#E1:182_2_2"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><a NAME="E5:182_2_2"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">e</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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a>  iff <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> )</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="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font> implies <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</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="E1:182_2_2_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</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="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">v9</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E3:182_2_2_1_1"/><span class="lab"><font color="Green" title="E94">A16</font></span>: 
<font color="Maroon" title="c10">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="c11">v9</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</a></span>;<br/><a NAME="E4:182_2_2_1_1"/>
<font color="Maroon" title="c11">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">u1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="182_2_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:182_2_2_1_1_1"/>
<font color="Maroon" title="c11">v9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">u1</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="E2:182_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:182_2_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E1:182_2_2"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="txt" href="#E3:182_2_2_1_1"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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><span class="kw">hence </span><a NAME="E5:182_2_2_1_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E4:182_2_2"><span class="lab"><font color="Green" title="E93">A15</font></span></a>, <a class="txt" href="#E3:182_2_2_1_1"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="ref" href="glib_000.html#D20" onmouseover="rs('glib_000/D20')" onmouseout="rh()">GLIB_000:def 20</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>

<span class="kw">reconsider </span><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c7">u1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:182"><span class="lab"><font color="Green" title="E89">A2</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:182_2_2_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e2</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="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesInOut()</a> </span><br/>

<a NAME="E4:182_2_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">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="c11">w</font>,<font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c10">e</font> is    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:182"><span class="lab"><font color="Green" title="E88">A1</font></span></a>, <a class="txt" href="#E4:182_2_2"><span class="lab"><font color="Green" title="E93">A15</font></span></a>, <a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:182_2_2_1"/>
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">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="ref" href="glib_000.html#T64" onmouseover="rs('glib_000/T64')" onmouseout="rh()">GLIB_000:64</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="E6:182_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="c9">e2</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/><span class="kw">hence </span><a NAME="E7:182_2_2"/>
<font color="Maroon" title="c4">v2</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="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</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>
