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

<b>let </b><font color="Maroon">IT1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>, <font color="Maroon">IT2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:55_1_2"/><i><font color="Green">E131</font></i>: 
( ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  holds <br/><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> holds <br/>( ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> implies <font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> ) &amp; ( not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> implies <font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> ) ) ) )
 <b>and </b><br/><a NAME="E2:55_1_2"/><i><font color="Green">E132</font></i>: 
( ( for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; not <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  holds <br/><font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> ) &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> holds <br/>( ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> implies <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> ) &amp; ( not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> implies <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:55_1_2_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_2_1_1_1"/>
(  not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  or <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_2_1_1_1_1"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><a NAME="E2:55_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:55_1_2_1_1_1_1"/>
<font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_2_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:55_1_2_1_1_1_2"/><i><font color="Green">E134</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> or  not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:55_1_2_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> &amp; <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:55_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_2_1_1_1_2_1_1_2"/>
not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:55_1_2_1_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> &amp; <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:55_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:55_1_2_1_1_1_2"/>
<font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:55_1_2_1"/>
<font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55_1_2"/>
<font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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