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

<span class="kw">let </span><font color="Maroon" title="c3">G2</font> be    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_1"/>
( <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> or  not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</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">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/><div class="add"><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font>, <font color="Olive" title="b3">v1</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c3">G2</font> &amp; <font color="Olive" title="b2">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b3">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c3">G2</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">e1</font>, <font color="Maroon" title="c5">e2</font>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</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="c4">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G2</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:75_1_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c3">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="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:75_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font> or <font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon" title="c9">f</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:75_1_1_1_1_1_1_1"/>
<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">G</font>
 <span class="kw">and </span><br/><a NAME="E4:75_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</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">G</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="E5:75_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> )
 <span class="kw">and </span><br/><a NAME="E6:75_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">and </span><br/><a NAME="E7:75_1_1_1_1_1_1_1"/>
 <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E8:75_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 not <font color="Maroon" title="c4">e1</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">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:75_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c4">e1</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">G</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="E2:75_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/>
<a NAME="E3:75_1_1_1_1_1_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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</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/>
<span class="kw">hence </span><a NAME="E4:75_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="glib_000.html#D18" onmouseover="rs('glib_000/D18')" onmouseout="rh()">GLIB_000:def 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><a NAME="E9:75_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
 not <font color="Maroon" title="c5">e2</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">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1_1_1_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:75_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c5">e2</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">G</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="E2:75_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/>
<a NAME="E3:75_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</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/>
<span class="kw">hence </span><a NAME="E4:75_1_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="glib_000.html#D18" onmouseover="rs('glib_000/D18')" onmouseout="rh()">GLIB_000:def 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><a NAME="E10:75_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E11:75_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E8:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E12:75_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E13:75_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c5">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E9:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e1</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E11:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">v1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E1:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E13:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
;<br/><span class="kw">hence </span><a NAME="E15:75_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E11:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E13:75_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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="75_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">v2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:75_1_1_1_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="glib_012.html#T17" target="_self" onmouseover="rs('glib_012/T17')" onmouseout="rh()">Th17</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e2</font>
 <span class="kw">by</span> <span class="lab"><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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:75_1_1_1_1"/>
<font color="Maroon" title="c3">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="75_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:75_1_1_1_2"/>
 not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</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">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> </span><br/><div class="add"><a NAME="E2:75_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:75_1_1_1_2"/>
<font color="Maroon" title="c3">G2</font> is  <a href="glib_000.html#V5" title="GLIB_000:attr.5">non-multi</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T89" onmouseover="rs('glib_000/T89')" onmouseout="rh()">GLIB_000:89</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>
