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

<span class="kw">let </span><font color="Maroon" title="c2">G2</font> be    <a href="glib_012.html#M2" title="GLIB_012:mode.2">DLGraphComplement</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">G2</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">w1</font>, <font color="Maroon" title="c4">w2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E2:101_1_1"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 for <font color="Olive" title="b1">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>  holds  not <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</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/>
<a NAME="E3:101_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:101_1_1_1"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:101_1_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_1"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E4:101_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:101_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:101_1_1"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_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 <font color="Maroon" title="c2">G2</font>  ex <font color="Olive" title="b3">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font> st <font color="Olive" title="b3">W</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="c6">v1</font>, <font color="Maroon" title="c7">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">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">W</font>,<font color="Olive">b<sub>2</sub></font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c8">u1</font> = <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c9">u2</font> = <font color="Maroon" title="c7">v2</font> as   <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="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_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="E1:101_1_1_2_1"/>
( (  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> &amp;  ex <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font> ) or ( (  for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> ) &amp;  ex <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font> ) or (  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font> ) ) or ( (  for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> ) &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font> ) ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
(  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> &amp;  ex <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">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">W</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="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c11">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
<font color="Maroon" title="c11">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a></span>;<br/><a NAME="E6:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E7:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
<font color="Maroon" title="c11">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E8:101_1_1_2_1_1"/>
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c12">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:101_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c12">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">W2</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>);<br/>
<a NAME="E3:101_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E4:101_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A10</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">W1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c6">v1</font>);<br/>
<a NAME="E5:101_1_1_2_1_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c6">v1</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E50">A7</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="E6:101_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c6">v1</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E53">A10</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
<font color="Maroon" title="c12">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E11:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E53">A12</font></span>: 
<font color="Maroon" title="c12">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E12:101_1_1_2_1_1"/>
 for <font color="Olive" title="b1">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c13">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:101_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
<font color="Maroon" title="c13">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c14">W1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c13">e4</font>,<font color="Maroon" title="c9">u2</font>);<br/>
<a NAME="E3:101_1_1_2_1_1_2"/>
<font color="Maroon" title="c13">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E4:101_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c13">e4</font>,<font color="Maroon" title="c9">u2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">W2</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c11">e2</font>,<font color="Maroon" title="c4">w2</font>);<br/>
<a NAME="E5:101_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c11">e2</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E51">A8</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="E6:101_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c13">e4</font>,<font color="Maroon" title="c9">u2</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c11">e2</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E14:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E54">A15</font></span>: 
<font color="Maroon" title="c13">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E15:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E55">A16</font></span>: 
<font color="Maroon" title="c13">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E54">A15</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">W1</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c15">W2</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">w1</font>);<br/><span class="kw">set </span><font color="Maroon" title="c16">W3</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c13">e4</font>,<font color="Maroon" title="c7">v2</font>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">W</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">w1</font>)</span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c13">e4</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c18">W</font> = <font color="Maroon" title="c17">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"> <font color="Maroon" title="c18">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><br/><a NAME="E17:101_1_1_2_1_1"/><span class="lab"><font color="Green" title="E56">A17</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E53">A12</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="E18:101_1_1_2_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E19:101_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">w1</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c3">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><a NAME="E20:101_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c12">e3</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c3">w1</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E56">A17</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/><a NAME="E21:101_1_1_2_1_1"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c13">e4</font>,<font color="Maroon" title="c7">v2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E55">A16</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><span class="kw">hence </span><a NAME="E22:101_1_1_2_1_1"/>
<font color="Maroon" title="c18">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:101_1_1_2_1_1"><span class="lab"><font color="Green" title="E57">A18</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</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="101_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
( (  for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> ) &amp;  ex <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">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">W</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="c10">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
<font color="Maroon" title="c10">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E4:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c10">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<font color="Maroon" title="c11">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E7:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<font color="Maroon" title="c11">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E8:101_1_1_2_1_2"/>
 for <font color="Olive" title="b1">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c12">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:101_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E52">A24</font></span>: 
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">W1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c9">u2</font>);<br/>
<a NAME="E3:101_1_1_2_1_2_1"/>
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E4:101_1_1_2_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A25</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c9">u2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">W2</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c10">e2</font>,<font color="Maroon" title="c4">w2</font>);<br/>
<a NAME="E5:101_1_1_2_1_2_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c10">e2</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E49">A21</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="E6:101_1_1_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c9">u2</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c10">e2</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E52">A26</font></span>: 
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E11:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A27</font></span>: 
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E52">A26</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">W1</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c3">w1</font>);<br/><span class="kw">set </span><font color="Maroon" title="c14">W2</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c7">v2</font>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c3">w1</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c16">W</font> = <font color="Maroon" title="c15">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"> <font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><br/><a NAME="E13:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E54">A28</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c3">w1</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E51">A23</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="E14:101_1_1_2_1_2"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c7">v2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E53">A27</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><span class="kw">hence </span><a NAME="E15:101_1_1_2_1_2"/>
<font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E54">A28</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</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="101_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E47">A29</font></span>: 
(  ex <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">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">W</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="c10">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E48">A30</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E4:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E49">A31</font></span>: 
<font color="Maroon" title="c10">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E48">A30</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E5:101_1_1_2_1_3"/>
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c11">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E2:101_1_1_2_1_3_1"/><span class="lab"><font color="Green" title="E50">A32</font></span>: 
<font color="Maroon" title="c11">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">W2</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c4">w2</font>);<br/>
<a NAME="E3:101_1_1_2_1_3_1"/>
<font color="Maroon" title="c11">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E50">A32</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/>
<a NAME="E4:101_1_1_2_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A33</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">W1</font> = <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c6">v1</font>);<br/>
<a NAME="E5:101_1_1_2_1_3_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c6">v1</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c6">v1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E49">A31</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="E6:101_1_1_2_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c10">e1</font>,<font color="Maroon" title="c6">v1</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_3_1"><span class="lab"><font color="Green" title="E51">A33</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:101_1_1_2_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1"><span class="lab"><font color="Green" title="E45">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<font color="Maroon" title="c11">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E8:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<font color="Maroon" title="c11">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E47">A29</font></span></a>, <a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E11:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">W1</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c4">w2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c14">W2</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c7">v2</font>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">W</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c16">W</font> = <font color="Maroon" title="c15">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"> <font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><br/><a NAME="E13:101_1_1_2_1_3"/><span class="lab"><font color="Green" title="E54">A38</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c11">e3</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E51">A35</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="E14:101_1_1_2_1_3"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c12">e4</font>,<font color="Maroon" title="c7">v2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><span class="kw">hence </span><a NAME="E15:101_1_1_2_1_3"/>
<font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:101_1_1_2_1_3"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</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="101_1_1_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_4"/><span class="lab"><font color="Green" title="E47">A39</font></span>: 
( (  for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c1">G</font> ) &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c4">w2</font>,<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">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">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">W</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="c10">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_2_1_4"/><span class="lab"><font color="Green" title="E48">A40</font></span>: 
<font color="Maroon" title="c10">e3</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c8">u1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E4:101_1_1_2_1_4"/><span class="lab"><font color="Green" title="E49">A41</font></span>: 
<font color="Maroon" title="c10">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E48">A40</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:101_1_1_2_1_4"/><span class="lab"><font color="Green" title="E50">A42</font></span>: 
<font color="Maroon" title="c11">e4</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">u2</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E47">A39</font></span></a>, <a class="ref" href="glib_012.html#D6" target="_self" onmouseover="rs('glib_012/D6')" onmouseout="rh()">Def6</a></span>;<br/><a NAME="E7:101_1_1_2_1_4"/><span class="lab"><font color="Green" title="E51">A43</font></span>: 
<font color="Maroon" title="c11">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c7">v2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E50">A42</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">W1</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c10">e3</font>,<font color="Maroon" title="c3">w1</font>);<br/><span class="kw">set </span><font color="Maroon" title="c13">W2</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">w2</font>);<br/><span class="kw">set </span><font color="Maroon" title="c14">W3</font> = <font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c11">e4</font>,<font color="Maroon" title="c7">v2</font>);<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">W</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c10">e3</font>,<font color="Maroon" title="c3">w1</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c11">e4</font>,<font color="Maroon" title="c7">v2</font>)</span>)</span> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c16">W</font> = <font color="Maroon" title="c15">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"> <font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font></span><br/><a NAME="E9:101_1_1_2_1_4"/><span class="lab"><font color="Green" title="E52">A44</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c10">e3</font>,<font color="Maroon" title="c3">w1</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c3">w1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E49">A41</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="E10:101_1_1_2_1_4"/>
<font color="Maroon" title="c5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E11:101_1_1_2_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">w2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c4">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><a NAME="E12:101_1_1_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A45</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c10">e3</font>,<font color="Maroon" title="c3">w1</font>)</span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c3">w1</font>,<font color="Maroon" title="c5">e</font>,<font color="Maroon" title="c4">w2</font>)</span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c4">w2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E52">A44</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</a></span>;<br/><a NAME="E13:101_1_1_2_1_4"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> (<font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c11">e4</font>,<font color="Maroon" title="c7">v2</font>) <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">w2</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E51">A43</font></span></a>, <a class="ref" href="glib_001.html#T15" onmouseover="rs('glib_001/T15')" onmouseout="rh()">GLIB_001:15</a></span>;<br/><span class="kw">hence </span><a NAME="E14:101_1_1_2_1_4"/>
<font color="Maroon" title="c16">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:101_1_1_2_1_4"><span class="lab"><font color="Green" title="E53">A45</font></span></a>, <a class="ref" href="glib_001.html#T31" onmouseover="rs('glib_001/T31')" onmouseout="rh()">GLIB_001:31</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="E7:101_1_1"/>
<font color="Maroon" title="c2">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/>


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