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

<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">V</font> holds <br/> ( <font color="Olive" title="b2">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  iff <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</font> holds <br/> ( <font color="Olive" title="b1">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  iff <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  iff <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span><br/>





<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  implies <font color="Maroon" title="c3">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:68_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c3">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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="c1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>  ex <font color="Olive" title="b3">P2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> st <font color="Olive" title="b3">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</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">P2</font>,<font color="Olive">b<sub>2</sub></font></span><br/><a NAME="E1:68_1_1"/>
( <font color="Maroon" title="c4">u</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c5">v</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T15" target="_self" onmouseover="rs('glib_012/T15')" onmouseout="rh()">Th15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G1</font><span class="kw"> such that </span><br/><a NAME="E3:68_1_1"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c6">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <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="c7">P1</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:68_1_1_1"/>
(  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G2</font> or  ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c3">G1</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">w</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T37" target="_self" onmouseover="rs('glib_012/T37')" onmouseout="rh()">Th37</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="68_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:68_1_1_1_1"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> is   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">P2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</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">P2</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="c8">P2</font> =  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c9">P2</font> = <font color="Maroon" title="c8">P2</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="c9">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><br/><a NAME="E3:68_1_1_1_1"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="glib_001.html#T160" onmouseover="rs('glib_001/T160')" onmouseout="rh()">GLIB_001:160</a></span>;<br/><span class="kw">hence </span><a NAME="E4:68_1_1_1_1"/>
<font color="Maroon" title="c9">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</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="68_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:68_1_1_1_2"/>
 ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b1">w</font>,<font color="Maroon" title="c3">G1</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">e</font>,<font color="Olive" title="b1">w</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">W2</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</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">W2</font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w</font>, <font color="Maroon" title="c9">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:68_1_1_1_2"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
( <font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c3">G1</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">e</font>,<font color="Maroon" title="c8">w</font>) )
 ;<br/><a NAME="E4:68_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
(  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">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">w</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">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="c8">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1_1_2"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><a NAME="E5:68_1_1_1_2"/>
 the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="glib_001.html#T160" onmouseover="rs('glib_001/T160')" onmouseout="rh()">GLIB_001:160</a></span>;<br/><a NAME="E6:68_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A5</font></span>: 
(  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">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="c4">u</font> &amp;  the   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c6">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="c5">v</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D23" onmouseover="rs('glib_001/D23')" onmouseout="rh()">GLIB_001:def 23</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">W2</font> = <font color="Maroon" title="c1">G2</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c4">u</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G2</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c11">W2</font> = <font color="Maroon" title="c10">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"> <font color="Maroon" title="c11">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><br/><span class="kw">thus </span><a NAME="E8:68_1_1_1_2"/>
<font color="Maroon" title="c11">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_1_1_2"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E6:68_1_1_1_2"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="glib_001.html#T13" onmouseover="rs('glib_001/T13')" onmouseout="rh()">GLIB_001:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:68_1"/>
<font color="Maroon" title="c1">G2</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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:68"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
<font color="Maroon" title="c1">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</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="c3">G1</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_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">u</font>, <font color="Olive" title="b2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font>  ex <font color="Olive" title="b3">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G1</font> st <font color="Olive" title="b3">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">v</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">W1</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G1</font> st <font color="Olive" title="b1">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><br/><a NAME="E1:68_2"/>
( <font color="Maroon" title="c4">u</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> &amp; <font color="Maroon" title="c5">v</font> is   <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="ref" href="glib_012.html#T15" target="_self" onmouseover="rs('glib_012/T15')" onmouseout="rh()">Th15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">W2</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="E3:68_2"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c6">W2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="glib_002.html#D1" onmouseover="rs('glib_002/D1')" onmouseout="rh()">GLIB_002:def 1</a></span>;<br/><a NAME="E4:68_2"/>
<font color="Maroon" title="c1">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c3">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T57" onmouseover="rs('glib_006/T57')" onmouseout="rh()">GLIB_006:57</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">W1</font> = <font color="Maroon" title="c6">W2</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T167" onmouseover="rs('glib_001/T167')" onmouseout="rh()">GLIB_001:167</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">W1</font> = <font color="Maroon" title="c7">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"> <font color="Maroon" title="c8">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font></span><br/><span class="kw">thus </span><a NAME="E6:68_2"/>
<font color="Maroon" title="c8">W1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">u</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_2"><span class="lab"><font color="Green" title="E33">A7</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>
<span class="kw">hence </span><a NAME="E4:68"/>
<font color="Maroon" title="c3">G1</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>; <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/>


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