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

<span class="kw">set </span><font color="Maroon" title="c2">D</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> ;<br/>
<div><span class="lab"><font color="Green" title="E103">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">W</font> be   <a href="glib_001.html#NM4" title="GLIB_001:NM.4">Trail</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:144_1_1_1"/><span class="lab"><font color="Green" title="E104">A2</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  holds <br/><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:144_1_1_1_1"/><span class="lab"><font color="Green" title="E105">A3</font></span>: 
<font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D27" target="_self" title="GLIB_001:def.27">Def27</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> implies <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:144_1_1_1_1"/><span class="lab"><font color="Green" title="E106">A4</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E3:144_1_1_1_1"/><span class="lab"><font color="Green" title="E107">A5</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:144_1_1_1_1"/><span class="lab"><font color="Green" title="E108">A6</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:144_1_1_1_1"/><span class="lab"><font color="Green" title="E109">A7</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E2:144_1_1_1_1"><span class="lab"><font color="Green" title="E106">A4</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E7:144_1_1_1_1"/><span class="lab"><font color="Green" title="E110">A8</font></span>: 
<font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E2:144_1_1_1_1"><span class="lab"><font color="Green" title="E106">A4</font></span></a>, <a class="txt" href="#E6:144_1_1_1_1"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></span>;<br/><a NAME="E8:144_1_1_1_1"/><span class="lab"><font color="Green" title="E111">A9</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_1"><span class="lab"><font color="Green" title="E107">A5</font></span></a></span>;<br/><a NAME="E9:144_1_1_1_1"/><span class="lab"><font color="Green" title="E112">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E2:144_1_1_1_1"><span class="lab"><font color="Green" title="E106">A4</font></span></a>, <a class="txt" href="#E6:144_1_1_1_1"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">a2</font>, <font color="Maroon" title="c11">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:144_1_1_1_1"/><span class="lab"><font color="Green" title="E113">A11</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_1"><span class="lab"><font color="Green" title="E107">A5</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E12:144_1_1_1_1"/><span class="lab"><font color="Green" title="E114">A12</font></span>: 
<font color="Maroon" title="c10">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_1"><span class="lab"><font color="Green" title="E107">A5</font></span></a>, <a class="txt" href="#E11:144_1_1_1_1"><span class="lab"><font color="Green" title="E113">A11</font></span></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></span>;<br/><a NAME="E13:144_1_1_1_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E2:144_1_1_1_1"><span class="lab"><font color="Green" title="E106">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A13</font></span>: <font color="Maroon" title="c9">b1</font> = 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:144_1_1_1_1"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c11">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:144_1_1_1_1"><span class="lab"><font color="Green" title="E108">A6</font></span></a>, <a class="txt" href="#E8:144_1_1_1_1"><span class="lab"><font color="Green" title="E111">A9</font></span></a>, <a class="txt" href="#E11:144_1_1_1_1"><span class="lab"><font color="Green" title="E113">A11</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>
;<br/><a NAME="E15:144_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_1"><span class="lab"><font color="Green" title="E107">A5</font></span></a>, <a class="txt" href="#E11:144_1_1_1_1"><span class="lab"><font color="Green" title="E113">A11</font></span></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E16:144_1_1_1_1"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:144_1_1_1_1"><span class="lab"><font color="Green" title="E109">A7</font></span></a>, <a class="txt" href="#E11:144_1_1_1_1"><span class="lab"><font color="Green" title="E113">A11</font></span></a>, <a class="txt" href="#E14:144_1_1_1_1"><span class="lab"><font color="Green" title="E115">A13</font></span></a>, <a class="txt" href="#E1:144_1_1_1_1"><span class="lab"><font color="Green" title="E105">A3</font></span></a>, <a class="txt" href="#E7:144_1_1_1_1"><span class="lab"><font color="Green" title="E110">A8</font></span></a>, <a class="txt" href="#E9:144_1_1_1_1"><span class="lab"><font color="Green" title="E112">A10</font></span></a>, <a class="txt" href="#E12:144_1_1_1_1"><span class="lab"><font color="Green" title="E114">A12</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:144_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A14</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> )</span><br/><span class="kw">assume </span><a NAME="E1:144_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:144_1_1_1_2"/><span class="lab"><font color="Green" title="E106">A15</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:144_1_1_1_2"/><span class="lab"><font color="Green" title="E107">A16</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:144_1_1_1_2"/><span class="lab"><font color="Green" title="E108">A17</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_2"><span class="lab"><font color="Green" title="E106">A15</font></span></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></span>;<br/><a NAME="E7:144_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_2"><span class="lab"><font color="Green" title="E106">A15</font></span></a>, <a class="txt" href="#E4:144_1_1_1_2"><span class="lab"><font color="Green" title="E107">A16</font></span></a></span>;<br/><a NAME="E8:144_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:144_1_1_1_2"><span class="lab"><font color="Green" title="E108">A17</font></span></a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></span>;<br/><a NAME="E9:144_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E3:144_1_1_1_2"><span class="lab"><font color="Green" title="E106">A15</font></span></a>, <a class="txt" href="#E6:144_1_1_1_2"><span class="lab"><font color="Green" title="E108">A17</font></span></a>, <a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:144_1_1_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:144_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E7:144_1_1_1"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:144_1_1_1"><span class="lab"><font color="Green" title="E104">A2</font></span></a>, <a class="txt" href="#E4:144_1_1_1"><span class="lab"><font color="Green" title="E105">A14</font></span></a>, <a class="ref" href="card_1.html#T26" title="CARD_1:th.26">CARD_1:26</a></span>;<br/><a NAME="E8:144_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T40" title="NAT_1:th.40">NAT_1:40</a></span>;<br/><a NAME="E9:144_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><a NAME="E10:144_1_1_1"/><span class="kw">then </span>
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E11:144_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E12:144_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D15" target="_self" title="GLIB_001:def.15">Def15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K22" title="GLIB_001:func.22">.allTrails()</a>  implies <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span>  )</span><br/><span class="kw">assume </span><a NAME="E1:144_1_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K22" title="GLIB_001:func.22">.allTrails()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">W</font> being   <a href="glib_001.html#NM4" title="GLIB_001:NM.4">Trail</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:144_1_1_2"/><span class="lab"><font color="Green" title="E104">A18</font></span>: 
<font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">e</font>
 ;<br/><a NAME="E4:144_1_1_2"/><span class="lab"><font color="Green" title="E105">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:144_1_1"><span class="lab"><font color="Green" title="E103">A1</font></span></a></span>;<br/><a NAME="E5:144_1_1_2"/>
<font color="Maroon" title="c4">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144_1_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><span class="kw">hence </span><a NAME="E6:144_1_1_2"/>
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:144_1_1_2"><span class="lab"><font color="Green" title="E104">A18</font></span></a>, <a class="txt" href="#E4:144_1_1_2"><span class="lab"><font color="Green" title="E105">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:144_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K22" title="GLIB_001:func.22">.allTrails()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K17" title="GLIB_000:func.17">.size()</a> </span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:144_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glib_001.html#K22" title="GLIB_001:func.22">.allTrails()</a>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a>, <a class="ref" href="graph_5.html#T4" title="GRAPH_5:th.4">GRAPH_5:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
