<?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#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G1</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font>  st <font color="Olive" title="b2">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> &amp;  not <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/>( <font color="Olive" title="b3">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Olive" title="b2">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <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="Olive" title="b3">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> <span class="p1">(<span class="default"><font color="Olive" title="b3">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Olive" title="b2">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</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">  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_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font>  st <font color="Olive" title="b1">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> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/>( <font color="Olive" title="b2">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Olive" title="b1">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <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="Olive" title="b2">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> <span class="p1">(<span class="default"><font color="Olive" title="b2">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Olive" title="b1">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>  st <font color="Maroon" title="c3">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> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/>( <font color="Olive" title="b1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <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="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Olive" title="b1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font> be    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c3">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> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> implies ( <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <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="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:91"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
( <font color="Maroon" title="c3">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> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <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="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> )</span><br/>

<a NAME="E2:91"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:91_1"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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/>

<span class="kw">set </span><font color="Maroon" title="c5">e</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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>;<br/>
<a NAME="E2:91_1"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>;<br/>
<a NAME="E3:91_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</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="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a>, <a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E60">A2</font></span></a></span>;<br/>
<a NAME="E4:91_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D30" onmouseover="rs('glib_000/D30')" onmouseout="rh()">GLIB_000:def 30</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:91_1"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
( <font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp;  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T102" onmouseover="rs('glib_000/T102')" onmouseout="rh()">GLIB_000:102</a></span>;<br/>
<a NAME="E7:91_1"/>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:91_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</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="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c1">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:91_1"><span class="lab"><font color="Green" title="E62">A4</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="E9:91_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</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="E3:91"/>
<font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</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">  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span></span><br/>

<a NAME="E4:91"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/>
<a NAME="E5:91"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:91"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<a NAME="E6:91"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="91_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">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="c4">G1</font> implies <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>;<br/>
<span class="kw">assume </span><a NAME="E1:91_2"/>
<font color="Maroon" title="c5">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="c4">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"> <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span></span><br/>

<a NAME="E2:91_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<a NAME="E3:91_2"/>
( <font color="Maroon" title="c5">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> or <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="91_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:91_2_1"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
 not <font color="Maroon" title="c5">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"> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:91_2_1"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
(  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c8">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="c4">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="c8">E</font> )
 <span class="kw">and </span><br/><a NAME="E4:91_2_1"/>
 for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_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="E1:91_2_1_1"/>
( ( <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="txt" href="#E3:91_2_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E2:91_2"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E1:91_2_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="glib_007.html#T51" target="_self" onmouseover="rs('glib_007/T51')" onmouseout="rh()">Th51</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:91_2_1_1_1"/>
( <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span><br/><div class="add"><a NAME="E2:91_2_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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> )
 <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="E3:91_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_2"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="glib_000.html#T17" onmouseover="rs('glib_000/T17')" onmouseout="rh()">GLIB_000:17</a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_2_1_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</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="glib_000.html#D30" onmouseover="rs('glib_000/D30')" onmouseout="rh()">GLIB_000:def 30</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="91_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:91_2_1_1_2"/>
( <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">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"> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span><br/><div class="add"><a NAME="E2:91_2_1_1_2"/><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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> )
 <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="E3:91_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_2"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="ref" href="glib_000.html#T17" onmouseover="rs('glib_000/T17')" onmouseout="rh()">GLIB_000:17</a></span>;<br/><span class="kw">hence </span><a NAME="E4:91_2_1_1_2"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</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="glib_000.html#D30" onmouseover="rs('glib_000/D30')" onmouseout="rh()">GLIB_000:def 30</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">hence </span><a NAME="E4:91_2"/>
<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0: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="E7:91"/><span class="kw">then </span>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:91"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:91"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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>
