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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  <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">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> <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="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a></span>)</span> holds <br/> ex <font color="Olive" title="b2">T</font> being   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Olive" title="b2">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="c2">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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a></span>)</span> implies  ex <font color="Olive" title="b1">T</font> being   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Olive" title="b1">T</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:213"/>
<font color="Maroon" title="c2">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">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a></span>)</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">  ex <font color="Olive" title="b1">T</font> being   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Olive" title="b1">T</font></span><br/>

<a NAME="E2:213"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A1</font></span>: 
( <font color="Maroon" title="c2">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">G</font> &amp;  not <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">T1</font> =  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="213_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="E1:213_1"/>
( <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> or  not <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="213_1_1"><span class="kw">suppose </span></a><a NAME="E1:213_1_1"/><span class="lab"><font color="Green" title="E129">A2</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <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">  ex <font color="Olive" title="b1">T</font> being   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Olive" title="b1">T</font></span><br/><div class="add"><span class="kw">take </span>
 the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <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="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font></span><br/><span class="kw">thus </span><a NAME="E2:213_1_1"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:213_1_1"><span class="lab"><font color="Green" title="E129">A2</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="213_1_2"><span class="kw">suppose </span></a><a NAME="E1:213_1_2"/><span class="lab"><font color="Green" title="E129">A3</font></span>: 
 not <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <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">  ex <font color="Olive" title="b1">T</font> being   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Olive" title="b1">T</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">v</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>;<br/><span class="kw">set </span><font color="Maroon" title="c5">w</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">G2</font> =  the    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>;<br/><a NAME="E2:213_1_2"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:213"><span class="lab"><font color="Green" title="E128">A1</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">G2</font> =  the    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c2">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="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font> as    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T50" onmouseover="rs('glibpre1/T50')" onmouseout="rh()">GLIBPRE1:50</a></span>;<br/><a NAME="E4:213_1_2"/>
( <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="c2">e</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</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="c2">e</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:213"><span class="lab"><font color="Green" title="E128">A1</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>;<br/><a NAME="E5:213_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A4</font></span>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">e</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</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="c2">e</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>;<br/><a NAME="E6:213_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A5</font></span>: 
(  not <font color="Maroon" title="c7">G2</font> is  <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</a>  &amp; (  for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">G2</font>  st <font color="Olive" title="b1">W1</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  &amp; <font color="Olive" title="b2">W2</font> is  <a href="glib_001.html#V8" title="GLIB_001:attr.8">Cycle-like</a>  holds <br/><font color="Olive" title="b1">W1</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:213_1_2"><span class="lab"><font color="Green" title="E129">A3</font></span></a>, <a class="ref" href="glibpre1.html#T51" onmouseover="rs('glibpre1/T51')" onmouseout="rh()">GLIBPRE1:51</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">G2</font><span class="kw"> such that </span><br/><a NAME="E8:213_1_2"/><span class="lab"><font color="Green" title="E132">A6</font></span>: 
<font color="Maroon" title="c8">W</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="ref" href="glib_002.html#D2" onmouseover="rs('glib_002/D2')" onmouseout="rh()">GLIB_002:def 2</a></span>;<br/><a NAME="E9:213_1_2"/><span class="lab"><font color="Green" title="E133">A7</font></span>: 
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <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="213_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:213_1_2_1"/>
 not <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">W1</font> = <font color="Maroon" title="c8">W</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:213_1_2"><span class="lab"><font color="Green" title="E130">A4</font></span></a>, <a class="ref" href="glib_006.html#T109" onmouseover="rs('glib_006/T109')" onmouseout="rh()">GLIB_006:109</a></span>;<br/>
<a NAME="E3:213_1_2_1"/>
<font color="Maroon" title="c9">W1</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="#E8:213_1_2"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="ref" href="glib_006.html#T24" onmouseover="rs('glib_006/T24')" onmouseout="rh()">GLIB_006:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:213_1_2_1"/>
contradiction
 <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><a NAME="E10:213_1_2"/>
<font color="Maroon" title="c7">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_009.html#T48" onmouseover="rs('glib_009/T48')" onmouseout="rh()">GLIB_009:48</a></span>;<br/><a NAME="E11:213_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:213"><span class="lab"><font color="Green" title="E128">A1</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:213_1_2"/><span class="lab"><font color="Green" title="E134">A8</font></span>: 
( <font color="Maroon" title="c9">e0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  &amp; <font color="Maroon" title="c9">e0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:213_1_2"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="txt" href="#E9:213_1_2"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="ref" href="glibpre1.html#T29" onmouseover="rs('glibpre1/T29')" onmouseout="rh()">GLIBPRE1:29</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">e0</font> = <font color="Maroon" title="c9">e0</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">T3</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">w1</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">w2</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font>;<br/><a NAME="E15:213_1_2"/><span class="lab"><font color="Green" title="E135">A9</font></span>: 
<font color="Maroon" title="c7">G2</font> is    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font>,<font color="Maroon" title="c10">e0</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:213_1_2"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="ref" href="glib_008.html#T37" onmouseover="rs('glib_008/T37')" onmouseout="rh()">GLIB_008:37</a></span>;<br/><a NAME="E16:213_1_2"/><span class="lab"><font color="Green" title="E136">A10</font></span>: 
<font color="Maroon" title="c10">e0</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font>,<font color="Maroon" title="c7">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:213_1_2"><span class="lab"><font color="Green" title="E134">A8</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="E17:213_1_2"/>
<font color="Maroon" title="c10">e0</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="c10">e0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E18:213_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">e0</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#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E19:213_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E137">A11</font></span>: 
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a>  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T51" onmouseover="rs('glib_000/T51')" onmouseout="rh()">GLIB_000:51</a></span>;<br/><a NAME="E20:213_1_2"/>
( <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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c7">G2</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c7">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:213_1_2"><span class="lab"><font color="Green" title="E136">A10</font></span></a>, <a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><a NAME="E21:213_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E138">A12</font></span>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="213_1_2_2"><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">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font>  ex <font color="Olive" title="b3">W3</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> st <font color="Olive" title="b3">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c14">v1</font>, <font color="Maroon" title="c15">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</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">  ex <font color="Olive" title="b1">W3</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W3</font>,<font color="Olive">b<sub>2</sub></font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c16">u1</font> = <font color="Maroon" title="c14">v1</font>, <font color="Maroon" title="c17">u2</font> = <font color="Maroon" title="c15">v2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c7">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">W2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c7">G2</font><span class="kw"> such that </span><br/><a NAME="E3:213_1_2_2"/><span class="lab"><font color="Green" title="E139">A13</font></span>: 
<font color="Maroon" title="c18">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c16">u1</font>,<font color="Maroon" title="c17">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">P2</font> =  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>;<br/><a NAME="E4:213_1_2_2"/><span class="lab"><font color="Green" title="E140">A14</font></span>: 
 the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c16">u1</font>,<font color="Maroon" title="c17">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:213_1_2_2"><span class="lab"><font color="Green" title="E139">A13</font></span></a>, <a class="ref" href="glib_001.html#T160" onmouseover="rs('glib_001/T160')" onmouseout="rh()">GLIB_001:160</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="213_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="E1:213_1_2_2_1"/>
(  not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  or <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="213_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:213_1_2_2_1_1"/>
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</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">  ex <font color="Olive" title="b1">W3</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W3</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">W3</font> =  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T172" onmouseover="rs('glib_001/T172')" onmouseout="rh()">GLIB_001:172</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c21">W3</font> = <font color="Maroon" title="c20">W3</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="c21">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font></span><br/><span class="kw">thus </span><a NAME="E3:213_1_2_2_1_1"/>
<font color="Maroon" title="c21">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:213_1_2_2"><span class="lab"><font color="Green" title="E140">A14</font></span></a>, <a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</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="213_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:213_1_2_2_1_2"/><span class="lab"><font color="Green" title="E141">A15</font></span>: 
<font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</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">  ex <font color="Olive" title="b1">W3</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W3</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c20">P4</font> being   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c7">G2</font><span class="kw"> such that </span><br/><a NAME="E3:213_1_2_2_1_2"/><span class="lab"><font color="Green" title="E142">A16</font></span>: 
( <font color="Maroon" title="c20">P4</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> &amp; <font color="Maroon" title="c20">P4</font> <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"><font color="Maroon" title="c8">W</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">e0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">and </span><br/><a NAME="E4:213_1_2_2_1_2"/>
(  not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  implies <font color="Maroon" title="c20">P4</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="#E8:213_1_2"><span class="lab"><font color="Green" title="E132">A6</font></span></a>, <a class="txt" href="#E13:213_1_2"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="txt" href="#E16:213_1_2"><span class="lab"><font color="Green" title="E136">A10</font></span></a>, <a class="ref" href="glibpre1.html#T32" onmouseover="rs('glibpre1/T32')" onmouseout="rh()">GLIBPRE1:32</a></span>;<br/><a NAME="E5:213_1_2_2_1_2"/>
<font color="Maroon" title="c10">e0</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="c10">e0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E6:213_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A17</font></span>: 
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E142">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E7:213_1_2_2_1_2"/><span class="lab"><font color="Green" title="E144">A18</font></span>: 
( <font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> &amp; <font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E142">A16</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/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="213_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="E1:213_1_2_2_1_2_1"/>
( <font color="Maroon" title="c7">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>,<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>) <a href="glib_006.html#R1" title="GLIB_006:pred.1">is_odd_substring_of</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>, <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  or <font color="Maroon" title="c7">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>,<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>) <a href="glib_006.html#R1" title="GLIB_006:pred.1">is_odd_substring_of</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>, <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:213_1_2"><span class="lab"><font color="Green" title="E136">A10</font></span></a>, <a class="txt" href="#E1:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E141">A15</font></span></a>, <a class="ref" href="glib_006.html#T28" onmouseover="rs('glib_006/T28')" onmouseout="rh()">GLIB_006:28</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="213_1_2_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:213_1_2_2_1_2_1_1"/><span class="lab"><font color="Green" title="E145">A19</font></span>: 
<font color="Maroon" title="c7">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>,<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>) <a href="glib_006.html#R1" title="GLIB_006:pred.1">is_odd_substring_of</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>, <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  ex <font color="Olive" title="b1">W3</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W3</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">W4</font> =  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<font color="Maroon" title="c20">P4</font>);<br/><a NAME="E2:213_1_2_2_1_2_1_1"/>
( <font color="Maroon" title="c10">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c7">G2</font> &amp; <font color="Maroon" title="c7">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span>) <a href="glib_006.html#R1" title="GLIB_006:pred.1">is_odd_substring_of</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>, <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:213_1_2"><span class="lab"><font color="Green" title="E136">A10</font></span></a>, <a class="txt" href="#E7:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E144">A18</font></span></a>, <a class="txt" href="#E1:213_1_2_2_1_2_1_1"><span class="lab"><font color="Green" title="E145">A19</font></span></a></span>;<br/><a NAME="E3:213_1_2_2_1_2_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<font color="Maroon" title="c20">P4</font>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E143">A17</font></span></a>, <a class="ref" href="glib_006.html#T40" onmouseover="rs('glib_006/T40')" onmouseout="rh()">GLIB_006:40</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">W3</font> =  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<font color="Maroon" title="c20">P4</font>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T172" onmouseover="rs('glib_001/T172')" onmouseout="rh()">GLIB_001:172</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c23">W3</font> = <font color="Maroon" title="c22">W3</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="c23">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font></span><br/><a NAME="E5:213_1_2_2_1_2_1_1"/>
 the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<font color="Maroon" title="c20">P4</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c16">u1</font>,<font color="Maroon" title="c17">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:213_1_2_2"><span class="lab"><font color="Green" title="E140">A14</font></span></a>, <a class="ref" href="glib_006.html#T47" onmouseover="rs('glib_006/T47')" onmouseout="rh()">GLIB_006:47</a></span>;<br/><span class="kw">hence </span><a NAME="E6:213_1_2_2_1_2_1_1"/>
<font color="Maroon" title="c23">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</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="213_1_2_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:213_1_2_2_1_2_1_2"/><span class="lab"><font color="Green" title="E145">A20</font></span>: 
<font color="Maroon" title="c7">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>,<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font></span>)</span>) <a href="glib_006.html#R1" title="GLIB_006:pred.1">is_odd_substring_of</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>, <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</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">  ex <font color="Olive" title="b1">W3</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> st <font color="Olive">b<sub>3</sub></font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">W3</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c21">W4</font> =  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>);<br/><a NAME="E2:213_1_2_2_1_2_1_2"/><span class="lab"><font color="Green" title="E146">A21</font></span>: 
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E143">A17</font></span></a>, <a class="ref" href="glib_001.html#T107" onmouseover="rs('glib_001/T107')" onmouseout="rh()">GLIB_001:107</a></span>;<br/><a NAME="E3:213_1_2_2_1_2_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></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> <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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></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> <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="c7">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">e0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:213_1_2_2_1_2"><span class="lab"><font color="Green" title="E144">A18</font></span></a>, <a class="ref" href="glib_001.html#T22" onmouseover="rs('glib_001/T22')" onmouseout="rh()">GLIB_001:22</a></span>;<br/><a NAME="E4:213_1_2_2_1_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c10">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c7">G2</font> &amp; <font color="Maroon" title="c7">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a></span>)</span>,<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a></span>)</span>) <a href="glib_006.html#R1" title="GLIB_006:pred.1">is_odd_substring_of</a>  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font>, <a href="ordinal1.html#K5" title="ORDINAL1:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:213_1_2"><span class="lab"><font color="Green" title="E136">A10</font></span></a>, <a class="txt" href="#E1:213_1_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E145">A20</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><a NAME="E5:213_1_2_2_1_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">e0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"> the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>)</span>)</span> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:213_1_2_2_1_2_1_2"><span class="lab"><font color="Green" title="E146">A21</font></span></a>, <a class="ref" href="glib_006.html#T40" onmouseover="rs('glib_006/T40')" onmouseout="rh()">GLIB_006:40</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">W3</font> =  the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>) as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T172" onmouseover="rs('glib_001/T172')" onmouseout="rh()">GLIB_001:172</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c23">W3</font> = <font color="Maroon" title="c22">W3</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="c23">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font></span><br/><a NAME="E7:213_1_2_2_1_2_1_2"/>
 the   <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>   <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c18">W2</font> <a href="glib_006.html#K4" title="GLIB_006:func.4">.replaceEdgeWith</a> (<font color="Maroon" title="c10">e0</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">P4</font> <a href="glib_001.html#K6" title="GLIB_001:func.6">.reverse()</a></span>)</span>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c16">u1</font>,<font color="Maroon" title="c17">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:213_1_2_2"><span class="lab"><font color="Green" title="E140">A14</font></span></a>, <a class="ref" href="glib_006.html#T47" onmouseover="rs('glib_006/T47')" onmouseout="rh()">GLIB_006:47</a></span>;<br/><span class="kw">hence </span><a NAME="E8:213_1_2_2_1_2_1_2"/>
<font color="Maroon" title="c23">W3</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T19" onmouseover="rs('glib_001/T19')" onmouseout="rh()">GLIB_001:19</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="E23:213_1_2"/><span class="kw">then </span>
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><a NAME="E24:213_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E139">A22</font></span>: 
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:213_1_2"><span class="lab"><font color="Green" title="E131">A5</font></span></a>, <a class="txt" href="#E15:213_1_2"><span class="lab"><font color="Green" title="E135">A9</font></span></a>, <a class="txt" href="#E19:213_1_2"><span class="lab"><font color="Green" title="E137">A11</font></span></a>, <a class="txt" href="#E21:213_1_2"><span class="lab"><font color="Green" title="E138">A12</font></span></a>, <a class="ref" href="glibpre1.html#T52" onmouseover="rs('glibpre1/T52')" onmouseout="rh()">GLIBPRE1:52</a></span>;<br/> <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> = 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c7">G2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>
<br/>.= 
 <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a>  the   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:213_1_2"><span class="lab"><font color="Green" title="E130">A4</font></span></a>, <a class="ref" href="glib_006.html#T102" onmouseover="rs('glib_006/T102')" onmouseout="rh()">GLIB_006:102</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">T3</font> =  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM8" title="GLIB_000:NM.8">removeEdge</a> of <font color="Maroon" title="c7">G2</font>,<font color="Maroon" title="c10">e0</font> as   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:213_1_2"><span class="lab"><font color="Green" title="E139">A22</font></span></a>, <a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a>, <a class="ref" href="glib_000.html#T43" onmouseover="rs('glib_000/T43')" onmouseout="rh()">GLIB_000:43</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c14">T3</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="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c14">T3</font></span><br/><a NAME="E27:213_1_2"/>
 not <font color="Maroon" title="c2">e</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="c10">e0</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="#E13:213_1_2"><span class="lab"><font color="Green" title="E134">A8</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E28:213_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c10">e0</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="#E9:213_1_2"><span class="lab"><font color="Green" title="E133">A7</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E29:213_1_2"/>
<font color="Maroon" title="c2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Edges_of</a> <font color="Maroon" title="c14">T3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T51" onmouseover="rs('glib_000/T51')" onmouseout="rh()">GLIB_000:51</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
