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<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> &amp; (  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  or  ex <font color="Olive" title="b4">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b5">w1</font>, <font color="Olive" title="b6">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b4">G3</font> st <br/>( <font color="Olive" title="b5">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b6">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b5">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">w2</font> ) ) holds <br/> not <font color="Olive" title="b3">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </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> &amp; (  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  or  ex <font color="Olive" title="b3">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b4">w1</font>, <font color="Olive" title="b5">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b3">G3</font> st <br/>( <font color="Olive" title="b4">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b5">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b4">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">w2</font> ) ) holds <br/> not <font color="Olive" title="b2">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </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> &amp; (  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  or  ex <font color="Olive" title="b2">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b3">w1</font>, <font color="Olive" title="b4">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b4">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">w2</font> ) ) holds <br/> not <font color="Olive" title="b1">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </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> &amp; (  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  or  ex <font color="Olive" title="b1">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w2</font> ) ) implies  not <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:111"/><span class="lab"><font color="Green" title="E69">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> )
 <span class="kw">and </span><br/><a NAME="E2:111"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
(  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  or  ex <font color="Olive" title="b1">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w2</font> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_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="E1:111_1"/>
(  not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  or  ex <font color="Olive" title="b1">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E70">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_1_1"/>
 not <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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">  not <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:111_1_1"/>
 not <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_1_2"/>
 ex <font color="Olive" title="b1">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font> ex <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b3">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Olive" title="b2">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">w2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">G3</font> being   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G2</font>, <font color="Maroon" title="c6">w1</font>, <font color="Maroon" title="c7">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c5">G3</font><span class="kw"> such that </span><br/><a NAME="E3:111_1_2"/><span class="lab"><font color="Green" title="E71">A3</font></span>: 
( <font color="Maroon" title="c6">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c7">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c6">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">w2</font> )
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">w1</font> = <font color="Maroon" title="c6">w1</font>, <font color="Maroon" title="c9">w2</font> = <font color="Maroon" title="c7">w2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E69">A1</font></span></a></span>;<br/><a NAME="E5:111_1_2"/>
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c5">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#T33" onmouseover="rs('glib_002/T33')" onmouseout="rh()">GLIB_002:33</a></span>;<br/><a NAME="E6:111_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">w2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c8">w1</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font><span class="kw"> such that </span><br/><a NAME="E8:111_1_2"/><span class="lab"><font color="Green" title="E72">A4</font></span>: 
<font color="Maroon" title="c10">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D5" onmouseover="rs('glib_002/D5')" onmouseout="rh()">GLIB_002:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">W1</font> = <font color="Maroon" title="c10">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c4">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T75" onmouseover="rs('glib_006/T75')" onmouseout="rh()">GLIB_006:75</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">P</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font>;<br/><a NAME="E10:111_1_2"/>
<font color="Maroon" title="c11">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_1_2"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</a></span>;<br/><a NAME="E11:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A5</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c9">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T160" onmouseover="rs('glib_001/T160')" onmouseout="rh()">GLIB_001:160</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E13:111_1_2"/><span class="lab"><font color="Green" title="E74">A6</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="c13">E</font> &amp; <font color="Maroon" title="c13">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="c13">E</font> )
 <span class="kw">and </span><br/><a NAME="E14:111_1_2"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
 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="c13">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">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e3</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">e3</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E69">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/><span class="kw">consider </span><font color="Maroon" title="c14">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E16:111_1_2"/><span class="lab"><font color="Green" title="E76">A8</font></span>: 
( <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E</font> &amp; <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E17:111_1_2"/>
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E14:111_1_2"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/><a NAME="E18:111_1_2"/><span class="lab"><font color="Green" title="E77">A9</font></span>: 
<font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:111_1_2"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E20:111_1_2"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
( <font color="Maroon" title="c15">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">E</font> &amp; <font color="Maroon" title="c15">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">w2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E21:111_1_2"/>
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">w2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c15">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E14:111_1_2"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c17">P3</font> = <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e1</font>;<br/><a NAME="E22:111_1_2"/><span class="lab"><font color="Green" title="E79">A11</font></span>: 
(  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:111_1_2"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/><a NAME="E23:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A12</font></span>: 
( <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font> &amp; <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:111_1_2"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E24:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A13</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c8">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:111_1_2"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="ref" href="glib_001.html#T63" onmouseover="rs('glib_001/T63')" onmouseout="rh()">GLIB_001:63</a></span>;<br/><a NAME="E25:111_1_2"/><span class="lab"><font color="Green" title="E82">A14</font></span>: 
 not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:111_1_2_1"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</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="E2:111_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T163" onmouseover="rs('glib_001/T163')" onmouseout="rh()">GLIB_001:163</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E3:111_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T98" onmouseover="rs('glib_001/T98')" onmouseout="rh()">GLIB_001:98</a></span>;<br/>
<a NAME="E4:111_1_2_1"/><span class="kw">then </span>
<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>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:111_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E69">A1</font></span></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="E26:111_1_2"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E22:111_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><a NAME="E27:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A15</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:111_1_2"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="txt" href="#E22:111_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="txt" href="#E25:111_1_2"><span class="lab"><font color="Green" title="E82">A14</font></span></a>, <a class="ref" href="glib_001.html#T151" onmouseover="rs('glib_001/T151')" onmouseout="rh()">GLIB_001:151</a></span>;<br/><a NAME="E28:111_1_2"/>
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E69">A1</font></span></a></span>;<br/><a NAME="E29:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A16</font></span>: 
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font> is  <a href="glib_001.html#NV5" title="GLIB_001:NV.5">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:111_1_2"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="glib_001.html#D24" onmouseover="rs('glib_001/D24')" onmouseout="rh()">GLIB_001:def 24</a></span>;<br/><a NAME="E30:111_1_2"/><span class="lab"><font color="Green" title="E85">A17</font></span>: 
 not <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:111_1_2_2"/><span class="lab"><font color="Green" title="E86">A18</font></span>: 
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</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="E2:111_1_2_2"/>
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:111_1_2"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="txt" href="#E22:111_1_2"><span class="lab"><font color="Green" title="E79">A11</font></span></a>, <a class="ref" href="glib_001.html#T111" onmouseover="rs('glib_001/T111')" onmouseout="rh()">GLIB_001:111</a></span>;<br/>
<a NAME="E3:111_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_2_2"><span class="lab"><font color="Green" title="E86">A18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:111_1_2_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  or <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_1_2_2_1_1"/>
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</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/><div class="add"><a NAME="E2:111_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T163" onmouseover="rs('glib_001/T163')" onmouseout="rh()">GLIB_001:163</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E3:111_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T110" onmouseover="rs('glib_001/T110')" onmouseout="rh()">GLIB_001:110</a></span>;<br/><a NAME="E4:111_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A19</font></span>: 
<font color="Maroon" title="c14">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">G2</font>
 ;<br/><a NAME="E5:111_1_2_2_1_1"/>
<font color="Maroon" title="c13">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:111_1_2"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:111_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:111_1_2"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E4:111_1_2_2_1_1"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="txt" href="glib_007.html#E35"><span class="lab"><font color="Green" title="E30">Lm7</font></span></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="111_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_1_2_2_1_2"/>
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">e2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:111_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">e2</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="E3:111_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c8">w1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:111_1_2"><span class="lab"><font color="Green" title="E77">A9</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_2_2_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:111_1_2_2_1_2_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> ) or ( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:111_1_2"><span class="lab"><font color="Green" title="E78">A10</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_2_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:111_1_2_2_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></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="111_1_2_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_1_2_2_1_2_1_2"/>
( <font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w2</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="E2:111_1_2_2_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E31:111_1_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> holds <br/><span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">w1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="111_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c18">n</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><a NAME="E2:111_1_2_3"/><span class="lab"><font color="Green" title="E86">A20</font></span>: 
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">n</font> &amp; <font color="Maroon" title="c18">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> &amp; <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</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="E3:111_1_2_3"/>
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:111_1_2"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="glib_001.html#D6" onmouseover="rs('glib_001/D6')" onmouseout="rh()">GLIB_001:def 6</a></span>;<br/>
<a NAME="E4:111_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c18">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_1_2_3"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="txt" href="#E27:111_1_2"><span class="lab"><font color="Green" title="E83">A15</font></span></a>, <a class="txt" href="glib_007.html#E89"><span class="lab"><font color="Green" title="E68">Lm16</font></span></a>, <a class="ref" href="glib_001.html#D28" onmouseover="rs('glib_001/D28')" onmouseout="rh()">GLIB_001:def 28</a></span>;<br/>
<a NAME="E5:111_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">w1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_1_2_3"><span class="lab"><font color="Green" title="E86">A20</font></span></a>, <a class="ref" href="glib_001.html#D7" onmouseover="rs('glib_001/D7')" onmouseout="rh()">GLIB_001:def 7</a></span>;<br/>
<a NAME="E6:111_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:111_1_2"><span class="lab"><font color="Green" title="E80">A12</font></span></a></span>;<br/>
<a NAME="E7:111_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_2"><span class="lab"><font color="Green" title="E71">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:111_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E69">A1</font></span></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="E32:111_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A21</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e1</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:111_1_2"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E23:111_1_2"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="txt" href="#E27:111_1_2"><span class="lab"><font color="Green" title="E83">A15</font></span></a>, <a class="txt" href="#E29:111_1_2"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="txt" href="#E30:111_1_2"><span class="lab"><font color="Green" title="E85">A17</font></span></a>, <a class="ref" href="glib_001.html#T150" onmouseover="rs('glib_001/T150')" onmouseout="rh()">GLIB_001:150</a></span>;<br/><a NAME="E33:111_1_2"/><span class="lab"><font color="Green" title="E87">A22</font></span>: 
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e1</font> is  <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:111_1_2"><span class="lab"><font color="Green" title="E81">A13</font></span></a>, <a class="ref" href="glib_001.html#T119" onmouseover="rs('glib_001/T119')" onmouseout="rh()">GLIB_001:119</a></span>;<br/><a NAME="E34:111_1_2"/>
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e1</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">V5</a>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:111_1_2"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E23:111_1_2"><span class="lab"><font color="Green" title="E80">A12</font></span></a>, <a class="ref" href="glib_001.html#T132" onmouseover="rs('glib_001/T132')" onmouseout="rh()">GLIB_001:132</a></span>;<br/><a NAME="E35:111_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"> the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c11">W1</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c15">e2</font></span>)</span> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c14">e1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:111_1_2"><span class="lab"><font color="Green" title="E86">A21</font></span></a>, <a class="txt" href="#E33:111_1_2"><span class="lab"><font color="Green" title="E87">A22</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E36:111_1_2"/>
 not <font color="Maroon" title="c4">G1</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 2</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>

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