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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">W1</font> being  <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">W1</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> holds <br/> ex <font color="Olive" title="b2">W2</font> being  <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Olive" title="b1">W1</font> st <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">W1</font> be   <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:136_1_1_1"/>
<font color="Maroon" title="c3">W1</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:136_1_1_1"/><b>then </b><i><font color="Green" title="E108">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E90"><i><font color="Green" title="E82">Lm53</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c4">W2</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c3">W1</font>;<br/><b>take </b><font color="Maroon" title="c5">W2</font> = <font color="Maroon" title="c4">W2</font>;<br/><a NAME="E4:136_1_1_1"/><i><font color="Green" title="E109">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_1"><i><font color="Green" title="E108">A1</font></i></a>, <a class="txt" href="glib_001.html#E115"><i><font color="Green" title="E107">Lm77</font></i></a></i>;<br/><a NAME="E5:136_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W2</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E6:136_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_1"><i><font color="Green" title="E109">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E7:136_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E97"><i><font color="Green" title="E89">Lm60</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:136_1_1_1"/><i><font color="Green" title="E110">A3</font></i>: 
<font color="Maroon" title="c5">W2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c6">v</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E98"><i><font color="Green" title="E90">Lm61</font></i></a></i>;<br/><b>thus </b><a NAME="E10:136_1_1_1"/>
<font color="Maroon" title="c5">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:136_1_1_1"><i><font color="Green" title="E110">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:136_1_1"/><b>then </b><i><font color="Green" title="E108">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2"/><i><font color="Green" title="E109">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>let </b><font color="Maroon" title="c4">W1</font> be   <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E2:136_1_1_2"/><i><font color="Green" title="E110">A6</font></i>: 
<font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><b>then </b><i><font color="Green" title="E111">A7</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W1</font> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="glib_001.html#E90"><i><font color="Green" title="E82">Lm53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2

;<br/><a NAME="E4:136_1_1_2"/><b>then </b><i><font color="Green" title="E112">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">W1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><b>set </b><font color="Maroon" title="c5">WA</font> = <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E5:136_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_2"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green" title="E55">Lm26</font></i></a></i>;<br/><a NAME="E6:136_1_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E90"><i><font color="Green" title="E82">Lm53</font></i></a></i>;<br/><a NAME="E7:136_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> </span>)</span>
 ;<br/><a NAME="E8:136_1_1_2"/><b>then </b><i><font color="Green" title="E113">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">WB</font> being   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E10:136_1_1_2"/><i><font color="Green" title="E114">A10</font></i>: 
<font color="Maroon" title="c6">WB</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2"><i><font color="Green" title="E109">A5</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c7">e</font> = <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">v</font> = <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/><a NAME="E11:136_1_1_2"/><i><font color="Green" title="E115">A11</font></i>: 
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_2"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="glib_001.html#E93"><i><font color="Green" title="E85">Lm56</font></i></a></i>;<br/><a NAME="E12:136_1_1_2"/><i><font color="Green" title="E116">A12</font></i>: 
<font color="Maroon" title="c6">WB</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D32" target="_self" title="GLIB_001:def.32">Def32</a></i>;<br/><a NAME="E13:136_1_1_2"/>
( not 1 is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> &amp; 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#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="ref" href="jordan12.html#T3" title="JORDAN12:th.3">JORDAN12:3</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E14:136_1_1_2"/><b>then </b><i><font color="Green" title="E117">A13</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></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">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_2"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>;<br/><a NAME="E15:136_1_1_2"/><b>then </b><i><font color="Green" title="E118">A14</font></i>: 
( <font color="Maroon" title="c6">WB</font> <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="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c6">WB</font> <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">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E12:136_1_1_2"><i><font color="Green" title="E116">A12</font></i></a>, <a class="ref" href="glib_001.html#D23" target="_self" title="GLIB_001:def.23">Def23</a></i>;<br/><a NAME="E16:136_1_1_2"/><b>then </b><i><font color="Green" title="E119">A15</font></i>: 
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:136_1_1_2"><i><font color="Green" title="E115">A11</font></i></a>, <a class="ref" href="glib_000.html#T19" title="GLIB_000:th.19">GLIB_000:19</a></i>;<br/><a NAME="E17:136_1_1_2"/><i><font color="Green" title="E120">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E84"><i><font color="Green" title="E76">Lm47</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1_2_2_1"/>
( <font color="Maroon" title="c6">WB</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> or  not <font color="Maroon" title="c6">WB</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_1"/>
<font color="Maroon" title="c6">WB</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c4">W1</font> <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="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E15:136_1_1_2"><i><font color="Green" title="E118">A14</font></i></a>, <a class="ref" href="glib_001.html#D24" target="_self" title="GLIB_001:def.24">Def24</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E121">A17</font></i>: 
<font color="Maroon" title="c4">W1</font> <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="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><b>set </b><font color="Maroon" title="c9">W2</font> = <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E4:136_1_1_2_2_1_1"/>
<font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2"><i><font color="Green" title="E111">A7</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_1"><i><font color="Green" title="E121">A17</font></i></a>, <a class="txt" href="glib_001.html#E73"><i><font color="Green" title="E65">Lm36</font></i></a></i>;<br/><b>then </b><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">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> 1,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2"><i><font color="Green" title="E111">A7</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2"><i><font color="Green" title="E112">A8</font></i></a>, <a class="txt" href="glib_001.html#E56"><i><font color="Green" title="E48">Lm19</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/><a NAME="E6:136_1_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">W2</font> = <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K20" title="GLIB_001:func.20">.remove</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c4">W1</font> <b>by </b><i><a class="txt" href="glib_001.html#E111"><i><font color="Green" title="E103">Lm74</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c11">W2</font> = <font color="Maroon" title="c10">W2</font>;<br/><b>thus </b><a NAME="E8:136_1_1_2_2_1_1"/>
<font color="Maroon" title="c11">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2"/><i><font color="Green" title="E121">A18</font></i>: 
not <font color="Maroon" title="c6">WB</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c9">esb</font> being    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2"/><i><font color="Green" title="E122">A19</font></i>: 
<font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D32" target="_self" title="GLIB_001:def.32">Def32</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2"/><i><font color="Green" title="E123">A20</font></i>: 
<font color="Maroon" title="c9">esb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><div><i><font color="Green" title="E124">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2"><i><font color="Green" title="E123">A20</font></i></a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E5:136_1_1_2_2_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E8:136_1_1_2"><i><font color="Green" title="E113">A9</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:136_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E125">A22</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2"/><i><font color="Green" title="E126">A23</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">esb</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E8:136_1_1_2_2_1_2"/><i><font color="Green" title="E127">A24</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2"><i><font color="Green" title="E122">A19</font></i></a>, <a class="ref" href="glib_001.html#D27" target="_self" title="GLIB_001:def.27">Def27</a></i>;<br/><a NAME="E9:136_1_1_2_2_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E10:136_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E128">A25</font></i>: 
( <font color="Maroon" title="c9">esb</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 <b>by </b><i><a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2"><i><font color="Green" title="E126">A23</font></i></a>, <a class="txt" href="glib_001.html#E8:136_1_1_2_2_1_2"><i><font color="Green" title="E127">A24</font></i></a>, <a class="ref" href="funct_1.html#T48" title="FUNCT_1:th.48">FUNCT_1:48</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1_2_2_1_2_2_1"/>
( <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or  not <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_1"/>
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c10">n</font> being  <a href="abian.html#NV3" title="ABIAN:NV.3">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E129">A26</font></i>: 
( <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">WB</font> &amp; <font color="Maroon" title="c6">WB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E86"><i><font color="Green" title="E78">Lm49</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">WB'</font> = <font color="Maroon" title="c6">WB</font> as  <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="glib_001.html#E10:136_1_1_2"><i><font color="Green" title="E114">A10</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c12">W2</font> = <font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_1"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 0</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c6">WB</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E129">A26</font></i></a>, <a class="txt" href="glib_001.html#E57"><i><font color="Green" title="E49">Lm20</font></i></a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E15:136_1_1_2"><i><font color="Green" title="E118">A14</font></i></a></i>;<br/><a NAME="E8:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2"><i><font color="Green" title="E111">A7</font></i></a></i>;<br/><a NAME="E9:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E130">A27</font></i>: 
<font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 ;<br/><a NAME="E10:136_1_1_2_2_1_2_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E129">A26</font></i></a>, <a class="txt" href="glib_001.html#E63"><i><font color="Green" title="E55">Lm26</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">naa1</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E12:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E131">A28</font></i>: 
( <font color="Maroon" title="c13">naa1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <b>by </b><i><a class="txt" href="glib_001.html#E83"><i><font color="Green" title="E75">Lm46</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c14">es</font> = <font color="Maroon" title="c9">esb</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c15">es</font> = <font color="Maroon" title="c9">esb</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#T29" title="GRAPH_2:th.29">GRAPH_2:29</a></i>;<br/><a NAME="E14:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E132">A29</font></i>: 
<font color="Maroon" title="c15">es</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><a NAME="E15:136_1_1_2_2_1_2_2_1_1"/>
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="c15">es</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E17:136_1_1_2"><i><font color="Green" title="E120">A16</font></i></a>, <a class="txt" href="glib_001.html#E14:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E132">A29</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E16:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c15">es</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">es</font> = <font color="Maroon" title="c15">es</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><a NAME="E18:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E133">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E84"><i><font color="Green" title="E76">Lm47</font></i></a></i>;<br/><a NAME="E19:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E134">A31</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">es</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E20:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E135">A32</font></i>: 
<font color="Maroon" title="c16">es</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_1"/><i><font color="Green" title="E136">A33</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_1_1_1"/>
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> or  not <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_1_1_2_2_1_2_2_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_1_1_1_2"/>
not <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_2_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_1_1_1"><i><font color="Green" title="E136">A33</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:136_1_1_2_2_1_2_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:136_1_1_2_2_1_2_2_1_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:136_1_1_2_2_1_2_2_1_1_1"/><i><font color="Green" title="E136">A34</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_2_1"/>
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font> or <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font> )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E136">A34</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_1_1_2_2_1_2_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E20:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E135">A32</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:136_1_1_2_2_1_2_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 ;<br/></div><b>end;</b></div><a NAME="E22:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">esb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">es</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E23:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E136">A35</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T13" title="RELAT_1:th.13">RELAT_1:13</a></i>;<br/><a NAME="E24:136_1_1_2_2_1_2_2_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/><a NAME="E25:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E137">A36</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E26:136_1_1_2_2_1_2_2_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E20:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E135">A32</font></i></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/><a NAME="E27:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E138">A37</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_2"/><i><font color="Green" title="E139">A38</font></i>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_1_2"><i><font color="Green" title="E139">A38</font></i></a></i>;<br/><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/></div><b>end;</b></div><a NAME="E29:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E30:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E139">A39</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E31:136_1_1_2_2_1_2_2_1_1"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c13">naa1</font>
 <b>by </b><i><a class="ref" href="pepin.html#T22" title="PEPIN:th.22">PEPIN:22</a></i>;<br/><a NAME="E32:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E140">A40</font></i>: 
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">naa1</font>
 <b>by </b><i><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><i><font color="Green" title="E141">A41</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2"><i><font color="Green" title="E122">A19</font></i></a>, <a class="ref" href="glib_001.html#T6" target="_self" title="GLIB_001:th.6">Th6</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_4"/>
<font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">naa1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</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="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E32:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E140">A40</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c13">naa1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</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="c6">WB</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#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E129">A26</font></i></a>, <a class="txt" href="glib_001.html#E12:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E131">A28</font></i></a>, <a class="ref" href="glib_001.html#D15" target="_self" title="GLIB_001:def.15">Def15</a></i>;<br/></div><b>end;</b></div><a NAME="E35:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E33:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E141">A41</font></i></a>, <a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/><a NAME="E36:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E142">A42</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">ses</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E38:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E143">A43</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">ses</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E36:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E142">A42</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E39:136_1_1_2_2_1_2_2_1_1"/>
<font color="Maroon" title="c17">ses</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E10:136_1_1_2_2_1_2"><i><font color="Green" title="E128">A25</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/><a NAME="E40:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E30:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E139">A39</font></i></a>, <a class="txt" href="glib_001.html#E38:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E143">A43</font></i></a>, <a class="ref" href="finseq_4.html#T77" title="FINSEQ_4:th.77">FINSEQ_4:77</a></i>;<br/><a NAME="E41:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c16">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><a NAME="E42:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E144">A44</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">es</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E12:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E131">A28</font></i></a>, <a class="ref" href="glib_001.html#T5" target="_self" title="GLIB_001:th.5">Th5</a></i>;<br/><a NAME="E43:136_1_1_2_2_1_2_2_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">esbes1</font> = <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">a</font>, <font color="Maroon" title="c20">b</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E145">A45</font></i>: 
( <font color="Maroon" title="c19">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font> &amp; <font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">esbes1</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c21">xa</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E146">A46</font></i>: 
( <font color="Maroon" title="c21">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">ses</font> &amp; <font color="Maroon" title="c17">ses</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">xa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E30:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E139">A39</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">xa</font> = <font color="Maroon" title="c21">xa</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E146">A46</font></i></a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E147">A47</font></i>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">xa</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E146">A46</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E148">A48</font></i>: 
( <font color="Maroon" title="c22">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> &amp; <font color="Maroon" title="c22">xa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E146">A46</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E149">A49</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">esbes1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E145">A45</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E8:136_1_1_2_2_1_2_2_1_1_5"/><b>then </b>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E9:136_1_1_2_2_1_2_2_1_1_5"/><b>then </b>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">xb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E150">A50</font></i>: 
( <font color="Maroon" title="c23">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">xb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">xb</font> = <font color="Maroon" title="c23">xb</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E11:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E150">A50</font></i></a></i>;<br/><a NAME="E13:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E151">A51</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">xa</font> &amp; <font color="Maroon" title="c22">xa</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 )
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E148">A48</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green" title="E152">A52</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">xb</font> &amp; <font color="Maroon" title="c24">xb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E33:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E141">A41</font></i></a>, <a class="txt" href="glib_001.html#E11:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E150">A50</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E15:136_1_1_2_2_1_2_2_1_1_5"/><b>then </b>
<font color="Maroon" title="c24">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2"><i><font color="Green" title="E122">A19</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E16:136_1_1_2_2_1_2_2_1_1_5"/><b>then </b>
<font color="Maroon" title="c24">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T6" target="_self" title="GLIB_001:th.6">Th6</a></i>;<br/><a NAME="E17:136_1_1_2_2_1_2_2_1_1_5"/><b>then </b><i><font color="Green" title="E153">A53</font></i>: 
<font color="Maroon" title="c24">xb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_5_1"/>
<font color="Maroon" title="c24">xb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">xa</font>
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c24">xb</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E13:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E151">A51</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1_5_1"/><b>then </b><i><font color="Green" title="E154">A54</font></i>: 
<font color="Maroon" title="c24">xb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E14:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E152">A52</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_1_5_1"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">xb</font>,<font color="Maroon" title="c20">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E11:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E150">A50</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_1_5_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">xb</font>,<font color="Maroon" title="c20">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ses</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_1_5_1"><i><font color="Green" title="E154">A54</font></i></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_1_5_1"/><b>then </b>
<font color="Maroon" title="c20">b</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="c17">ses</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">b</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">esbes1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E149">A49</font></i></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></i>;<br/><b>hence </b><a NAME="E8:136_1_1_2_2_1_2_2_1_1_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E149">A49</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:136_1_1_2_2_1_2_2_1_1_5"/>
<font color="Maroon" title="c19">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">b</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E147">A47</font></i></a>, <a class="txt" href="glib_001.html#E11:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E150">A50</font></i></a>, <a class="txt" href="glib_001.html#E13:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E151">A51</font></i></a>, <a class="txt" href="glib_001.html#E17:136_1_1_2_2_1_2_2_1_1_5"><i><font color="Green" title="E153">A53</font></i></a>, <a class="ref" href="finseq_1.html#D13" title="FINSEQ_1:def.13">FINSEQ_1:def 13</a></i>;<br/></div><b>end;</b></div><a NAME="E46:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E145">A55</font></i>: 
 <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E23:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E136">A35</font></i></a>, <a class="txt" href="glib_001.html#E25:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E137">A36</font></i></a>, <a class="txt" href="glib_001.html#E27:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E138">A37</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>;<br/><a NAME="E47:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green" title="E146">A56</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <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="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2"><i><font color="Green" title="E122">A19</font></i></a>, <a class="txt" href="glib_001.html#E18:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E133">A30</font></i></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green" title="E147">A57</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">x</font> &amp; <font color="Maroon" title="c19">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> )
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b><i><font color="Green" title="E148">A58</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <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="c19">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>then </b><i><font color="Green" title="E149">A59</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <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="c19">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
<b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2"><i><font color="Green" title="E122">A19</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a>, <a class="txt" href="glib_001.html#E18:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E133">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2"><i><font color="Green" title="E126">A23</font></i></a>, <a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E148">A58</font></i></a>, <a class="txt" href="glib_001.html#E47:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E146">A56</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green" title="E150">A60</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">es</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E42:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E144">A44</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E147">A57</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T6" target="_self" title="GLIB_001:th.6">Th6</a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b><i><font color="Green" title="E151">A61</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E46:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E145">A55</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E8:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green" title="E152">A62</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2"><i><font color="Green" title="E126">A23</font></i></a>, <a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E148">A58</font></i></a>, <a class="txt" href="glib_001.html#E47:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E146">A56</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E9:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b><i><font color="Green" title="E153">A63</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E10:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green" title="E154">A64</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">naa1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E147">A57</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="glib_001.html#E12:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E131">A28</font></i></a></i>;<br/><a NAME="E11:136_1_1_2_2_1_2_2_1_1_6"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E8:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E152">A62</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E12:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">ses</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E10:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E154">A64</font></i></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></i>;<br/><a NAME="E13:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</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="c17">ses</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><a NAME="E14:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E9:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E153">A63</font></i></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></i>;<br/><a NAME="E15:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b>
<font color="Maroon" title="c16">es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E151">A61</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><b>hence </b><a NAME="E16:136_1_1_2_2_1_2_2_1_1_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <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="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E149">A59</font></i></a>, <a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_1_6"><i><font color="Green" title="E150">A60</font></i></a>, <a class="txt" href="glib_001.html#E19:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E134">A31</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/></div><b>end;</b></div><a NAME="E49:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c16">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E42:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E144">A44</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">W2</font> = <font color="Maroon" title="c11">WB'</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c10">n</font> as   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c4">W1</font> <b>by </b><i><a class="txt" href="glib_001.html#E9:136_1_1_2_2_1_2_2_1_1"><i><font color="Green" title="E130">A27</font></i></a>, <a class="ref" href="glib_001.html#D32" target="_self" title="GLIB_001:def.32">Def32</a></i>;<br/><b>take </b><font color="Maroon" title="c20">W2</font> = <font color="Maroon" title="c19">W2</font>;<br/><b>thus </b><a NAME="E51:136_1_1_2_2_1_2_2_1_1"/>
<font color="Maroon" title="c20">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E129">A65</font></i>: 
not <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c10">W2</font> = <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_2"/>
( <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> &amp; <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1,<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E10:136_1_1_2"><i><font color="Green" title="E114">A10</font></i></a>, <a class="txt" href="glib_001.html#E11:136_1_1_2"><i><font color="Green" title="E115">A11</font></i></a>, <a class="txt" href="glib_001.html#E12:136_1_1_2"><i><font color="Green" title="E116">A12</font></i></a>, <a class="txt" href="glib_001.html#E14:136_1_1_2"><i><font color="Green" title="E117">A13</font></i></a>, <a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 ;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E130">A66</font></i>: 
<font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> ,<font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2"><i><font color="Green" title="E111">A7</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c11">es</font> = <font color="Maroon" title="c9">esb</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E131">A67</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5" title="CQC_LANG:th.5">CQC_LANG:5</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E132">A68</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_1"/><i><font color="Green" title="E133">A69</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_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="E1:136_1_1_2_2_1_2_2_1_2_1_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> or <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E132">A68</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_1"><i><font color="Green" title="E133">A69</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_2_2_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E134">A70</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2"><i><font color="Green" title="E124">A21</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">x'</font> = <font color="Maroon" title="c12">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_1_1_1_1"/><i><font color="Green" title="E135">A71</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">x'</font> &amp; <font color="Maroon" title="c13">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_1_1_1_1"><i><font color="Green" title="E134">A70</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E6:136_1_1_2_2_1_2_2_1_2_1_1_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2_1_1_1_1"><i><font color="Green" title="E135">A71</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_2_2_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E134">A72</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_2_1_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_1_1_1_2"><i><font color="Green" title="E134">A72</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:136_1_1_2_2_1_2_2_1_2_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/></div><b>end;</b></div><a NAME="E8:136_1_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E133">A73</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">esb</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">es</font> = <font color="Maroon" title="c9">esb</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> as   <a href="finseq_1.html#NM4" title="FINSEQ_1:NM.4">FinSubsequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D12" title="FINSEQ_1:def.12">FINSEQ_1:def 12</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_2"/><i><font color="Green" title="E134">A74</font></i>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">es</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">x</font>, <font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_2"/><i><font color="Green" title="E135">A75</font></i>: 
<font color="Maroon" title="c13">z</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="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_2"/><i><font color="Green" title="E136">A76</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">es</font> &amp; <font color="Maroon" title="c12">es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E134">A74</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E135">A75</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1_2_2_1_2_2_1_2_2_1_1"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> or  not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><i><font color="Green" title="E137">A77</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E138">A78</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E132">A68</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E136">A76</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><i><font color="Green" title="E139">A79</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E137">A77</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E140">A80</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E138">A78</font></i></a>, <a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">x'</font> = <font color="Maroon" title="c14">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E137">A77</font></i></a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><i><font color="Green" title="E141">A81</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">x'</font> &amp; <font color="Maroon" title="c16">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E139">A79</font></i></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="txt" href="glib_001.html#E2:136_1_1_2"><i><font color="Green" title="E110">A6</font></i></a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E142">A82</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</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="c14">x</font> = 
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E139">A79</font></i></a>, <a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E141">A81</font></i></a>, <a class="ref" href="glib_001.html#D15" target="_self" title="GLIB_001:def.15">Def15</a></i>
<br/>.= 
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

;<br/><b>hence </b><a NAME="E9:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E135">A75</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E140">A80</font></i></a>, <a class="txt" href="glib_001.html#E7:136_1_1_2_2_1_2_2_1_2_2_1_1_1"><i><font color="Green" title="E142">A82</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2_2_1_1_2"/><i><font color="Green" title="E137">A83</font></i>: 
not <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_2_2_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E138">A84</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E132">A68</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E136">A76</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E132">A68</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E136">A76</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_2_1_1_2"><i><font color="Green" title="E137">A83</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">esb</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_2_1_1_2"><i><font color="Green" title="E138">A84</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_2_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x</font>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2"><i><font color="Green" title="E123">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E6:136_1_1_2_2_1_2_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="txt" href="glib_001.html#E17:136_1_1_2"><i><font color="Green" title="E120">A16</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_2"><i><font color="Green" title="E135">A75</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:136_1_1_2_2_1_2_2_1_2_2"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 ;<br/></div><b>end;</b></div><a NAME="E11:136_1_1_2_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c12">es</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">es</font> = <font color="Maroon" title="c12">es</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c4">W1</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#D5" title="GRAPH_2:def.5">GRAPH_2:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_3"/>
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_3"/><i><font color="Green" title="E134">A85</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_3"/><i><font color="Green" title="E135">A86</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> &amp; <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_3"><i><font color="Green" title="E134">A85</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_2_3"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2_3"><i><font color="Green" title="E135">A86</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:136_1_1_2_2_1_2_2_1_2_3"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E14:136_1_1_2_2_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E134">A87</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E15:136_1_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E135">A88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E16:136_1_1_2"><i><font color="Green" title="E119">A15</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2"><i><font color="Green" title="E122">A19</font></i></a>, <a class="txt" href="glib_001.html#E85"><i><font color="Green" title="E77">Lm48</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E136">A89</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> = 
<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"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<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"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="glib_001.html#T5" target="_self" title="GLIB_001:th.5">Th5</a></i>
;<br/><i><font color="Green" title="E137">A90</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">es</font>
<b>by </b><i><a class="ref" href="glib_001.html#T5" target="_self" title="GLIB_001:th.5">Th5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E14:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E134">A87</font></i></a>, <a class="ref" href="pre_circ.html#T27" title="PRE_CIRC:th.27">PRE_CIRC:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T45" title="CQC_LANG:th.45">CQC_LANG:45</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E16:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E136">A89</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
;<br/><a NAME="E18:136_1_1_2_2_1_2_2_1_2"/><i><font color="Green" title="E138">A91</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E132">A68</font></i></a>, <a class="txt" href="glib_001.html#E8:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E133">A73</font></i></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">m</font>, <font color="Maroon" title="c15">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_6"/><i><font color="Green" title="E139">A92</font></i>: 
( <font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> &amp; <font color="Maroon" title="c15">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_2_6"/><b>then </b><i><font color="Green" title="E140">A93</font></i>: 
<font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_6"/>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_6"><i><font color="Green" title="E139">A92</font></i></a></i>;<br/><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_2_6"/>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_6"><i><font color="Green" title="E140">A93</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E139">A94</font></i>:  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">es</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E132">A68</font></i></a>, <a class="txt" href="glib_001.html#E18:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E138">A91</font></i></a>, <a class="ref" href="finseq_3.html#T48" title="FINSEQ_3:th.48">FINSEQ_3:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_3.html#T50" title="FINSEQ_3:th.50">FINSEQ_3:50</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_8"/><i><font color="Green" title="E140">A95</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> )
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_2_8"/><b>then </b><i><font color="Green" title="E141">A96</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_8"/><i><font color="Green" title="E142">A97</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E140">A95</font></i></a>, <a class="txt" href="glib_001.html#E17:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E137">A90</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_8"/>
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">es</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_2_8"/><b>then </b><i><font color="Green" title="E143">A98</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E142">A97</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_2_8"/><i><font color="Green" title="E144">A99</font></i>: 
 <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">esb</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14" title="FINSEQ_1:def.14">FINSEQ_1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:136_1_1_2_2_1_2_2_1_2_8_1_1"/>
( <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span> or  ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">m</font> ) )
 </span><b>by </b><i><a class="txt" href="glib_001.html#E15:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E135">A88</font></i></a>, <a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E141">A96</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_8_1_1_1"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2_8_1_1_1"/><i><font color="Green" title="E145">A100</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E146">A101</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="c14">n</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
<b>by </b><i><a class="txt" href="glib_001.html#E15:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E135">A88</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c9">esb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E144">A99</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_8_1_1_1"><i><font color="Green" title="E145">A100</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_8_1_1_1"/><i><font color="Green" title="E147">A102</font></i>: 
( <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E144">A99</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2_8_1_1_1"><i><font color="Green" title="E145">A100</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_8_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E20:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E139">A94</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:136_1_1_2_2_1_2_2_1_2_8_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E2:136_1_1_2_2_1_2_2_1_2_8_1_1_1"><i><font color="Green" title="E146">A101</font></i></a>, <a class="txt" href="glib_001.html#E14:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E134">A87</font></i></a>, <a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E143">A98</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_8_1_1_1"><i><font color="Green" title="E147">A102</font></i></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="136_1_1_2_2_1_2_2_1_2_8_1_1_2"><b>suppose </b></a><a NAME="E1:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/>
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Olive" title="b1">m</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c15">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/><i><font color="Green" title="E145">A103</font></i>: 
( <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">m</font> )
 ;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/>
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default">1</span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E145">A103</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/><b>then </b><i><font color="Green" title="E146">A104</font></i>: 
<font color="Maroon" title="c15">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E147">A105</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="c14">n</font> = 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="glib_001.html#E15:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E135">A88</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E145">A103</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>
;<br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2"><i><font color="Green" title="E125">A22</font></i></a>, <a class="ref" href="finseq_3.html#T44" title="FINSEQ_3:th.44">FINSEQ_3:44</a></i>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">esb</font>
<b>by </b><i><a class="txt" href="glib_001.html#E3"><i><font color="Green" title="E3">Lm2</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c9">esb</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#T5" target="_self" title="GLIB_001:th.5">Th5</a></i>
;<br/><a NAME="E8:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/><b>then </b><i><font color="Green" title="E148">A106</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">es</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2_8"><i><font color="Green" title="E143">A98</font></i></a>, <a class="txt" href="glib_001.html#E20:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E139">A94</font></i></a>, <a class="txt" href="glib_001.html#E3:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E145">A103</font></i></a>, <a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E146">A104</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>;<br/><a NAME="E9:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/><i><font color="Green" title="E149">A107</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E131">A67</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">esb</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E8:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E148">A106</font></i></a>, <a class="txt" href="glib_001.html#E9:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E149">A107</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6" title="CQC_LANG:th.6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E12:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E6:136_1_1_2_2_1_2_2_1_2_8_1_1_2"><i><font color="Green" title="E147">A105</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E8:136_1_1_2_2_1_2_2_1_2_8"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <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="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font>
 ;<br/></div><b>end;</b></div><a NAME="E22:136_1_1_2_2_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K15" title="FINSEQ_1:func.15">Seq</a> <font color="Maroon" title="c13">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E17:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E137">A90</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">W2</font> = <font color="Maroon" title="c6">WB</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> as   <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c4">W1</font> <b>by </b><i><a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E129">A65</font></i></a>, <a class="txt" href="glib_001.html#E16:136_1_1_2"><i><font color="Green" title="E119">A15</font></i></a>, <a class="txt" href="glib_001.html#E1:136_1_1_2_2_1_2"><i><font color="Green" title="E121">A18</font></i></a>, <a class="txt" href="glib_001.html#E110"><i><font color="Green" title="E102">Lm73</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1_2_2_1_2_2_1_2"><i><font color="Green" title="E130">A66</font></i></a>, <a class="ref" href="glib_001.html#D32" target="_self" title="GLIB_001:def.32">Def32</a></i>;<br/><b>take </b><font color="Maroon" title="c15">W2</font> = <font color="Maroon" title="c14">W2</font>;<br/><b>thus </b><a NAME="E24:136_1_1_2_2_1_2_2_1_2"/>
<font color="Maroon" title="c15">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E10:136_1_1_2"><i><font color="Green" title="E114">A10</font></i></a>, <a class="txt" href="glib_001.html#E11:136_1_1_2"><i><font color="Green" title="E115">A11</font></i></a>, <a class="txt" href="glib_001.html#E12:136_1_1_2"><i><font color="Green" title="E116">A12</font></i></a>, <a class="txt" href="glib_001.html#E14:136_1_1_2"><i><font color="Green" title="E117">A13</font></i></a>, <a class="txt" href="glib_001.html#E94"><i><font color="Green" title="E86">Lm57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:136_1_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">W2</font> being  <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c4">W1</font> st <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E19:136_1_1_2"/>
 ex <font color="Olive" title="b1">W2</font> being  <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c4">W1</font> st <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:136_1_1"/><b>then </b><i><font color="Green" title="E109">A108</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<a NAME="E5:136_1_1"/><i><font color="Green" title="E110">A109</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_001.html#E2:136_1_1"><i><font color="Green" title="E108">A4</font></i></a>, <a class="txt" href="glib_001.html#E4:136_1_1"><i><font color="Green" title="E109">A108</font></i></a>);<br/></i>
<a NAME="E6:136_1_1"/>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K15" title="GLIB_001:func.15">.length()</a> 
 
;<br/>
<b>hence </b><a NAME="E7:136_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="glib_001.html#NM11" title="GLIB_001:NM.11">Path</a> of <font color="Maroon" title="c2">W</font> st <font color="Olive">b<sub>1</sub></font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a>
 <b>by </b><i><a class="txt" href="glib_001.html#E5:136_1_1"><i><font color="Green" title="E110">A109</font></i></a></i>;<br/>


</div>
