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

<b>let </b><font color="Maroon">G</font> be   <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>, <font color="Maroon">W2</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/>



<b>set </b><font color="Maroon">W3</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font>;<br/>
<b>set </b><font color="Maroon">W4</font> = <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span>;<br/>
<b>assume </b><a NAME="E1:164"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3">.first()</a> 
 ;<br/>

<a NAME="E2:164"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K12">.edgeSeq()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#T86">GLIB_001:86</a></i>;<br/>
<a NAME="E3:164"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<b>then </b><i><font color="Green">E32</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="164_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:164_2"/><i><font color="Green">E51</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> )
 ;<br/><b>then </b><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_003.html#E4:164"><i><font color="Green">E32</font></i></a>, <a class="ref" href="glib_001.html#T86">GLIB_001:86</a></i>
;<br/><a NAME="E3:164_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="164_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:164_2_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_2_2_1_1"><b>suppose </b></a><a NAME="E1:164_2_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span>
 ;<br/><div class="add"><a NAME="E2:164_2_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E59</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E4:164_2_2_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E5:164_2_2_1_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E6:164_2_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="164_2_2_1_2"><b>suppose </b></a><a NAME="E1:164_2_2_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:164_2_2_1_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 ;<br/><a NAME="E4:164_2_2_1_2"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, </i>
;<br/><a NAME="E6:164_2_2_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:164_2_2_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E8:164_2_2_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:164_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:164_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:164"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K8" target="_self">.weightSeq()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K8" target="_self">.weightSeq()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


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