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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c1">G</font> holds <br/>( <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="Olive" title="b2">x</font>,<font color="Olive" title="b1">e</font>,<font color="Olive" title="b3">y</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> &amp; <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="Olive" title="b2">x</font>,<font color="Olive" title="b1">e</font>,<font color="Olive" title="b3">y</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y</font> &amp; <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">e</font>,<font color="Olive" title="b3">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> )</span><br/><b>let </b><font color="Maroon" title="c2">e</font>, <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c1">G</font> implies ( <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> ) )</span><br/>







<b>set </b><font color="Maroon" title="c5">W</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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font>;<br/>
<b>assume </b><a NAME="E1:157"/>
<font color="Maroon" title="c2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> )</span><br/>

<a NAME="E2:157"/><b>then </b><i><font color="Green" title="E109">A1</font></i>: 
<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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D5" target="_self" title="GLIB_001:def.5">Def5</a></i>;<br/>
<a NAME="E3:157"/><b>then </b><i><font color="Green" title="E110">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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="c3">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>;<br/>
<b>thus </b><a NAME="E4:157"/><i><font color="Green" title="E111">A3</font></i>: 
<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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E2:157"><i><font color="Green" title="E109">A1</font></i></a>, <a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> &amp; <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> )</span><br/>

<b>thus </b><a NAME="E5:157"/>
<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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E3:157"><i><font color="Green" title="E110">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><br/>

<b>hence </b><a NAME="E6:157"/>
<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">x</font>,<font color="Maroon" title="c2">e</font>,<font color="Maroon" title="c4">y</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E4:157"><i><font color="Green" title="E111">A3</font></i></a>, <a class="ref" href="glib_001.html#D23" target="_self" title="GLIB_001:def.23">Def23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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