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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">W1</font> being  <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">G</font>  st <font color="Olive">W1</font> <a href="glib_001.html#K15" target="_self">.length()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">W2</font> being  <a href="glib_001.html#NM11">Path</a> of <font color="Olive">W1</font> st <font color="Olive">W2</font> is <a href="glib_001.html#V2" target="_self">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">W1</font> be   <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:136_1_1_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K15" target="_self">.length()</a>  <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:136_1_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">W2</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><a NAME="E4:136_1_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="glib_001.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E5:136_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E6:136_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:136_1_1_1"/><b>then </b>
<font color="Maroon">W2</font> is <a href="glib_001.html#V3" target="_self">trivial</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">v</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:136_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">W2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K1" target="_self">.walkOf</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E10:136_1_1_1"/>
<font color="Maroon">W2</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E2:136_1_1"/><b>then </b><i><font color="Green">E43</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">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><b>let </b><font color="Maroon">W1</font> be   <a href="glib_001.html#NM6">DWalk</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E2:136_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K15" target="_self">.length()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><b>then </b><i><font color="Green">E46</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font> = 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2

;<br/><a NAME="E4:136_1_1_2"/><b>then </b><i><font color="Green">E47</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>set </b><font color="Maroon">WA</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E5:136_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, </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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K15" target="_self">.length()</a> </span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E7:136_1_1_2"/><b>then </b>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K15" target="_self">.length()</a> </span>)</span>
 ;<br/><a NAME="E8:136_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K15" target="_self">.length()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><b>then consider </b><font color="Maroon">WB</font> being   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span><b> such that </b><br/><a NAME="E10:136_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">WB</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">e</font> = <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">v</font> = <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>;<br/><a NAME="E11:136_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><a NAME="E12:136_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">WB</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i/>;<br/><a NAME="E13:136_1_1_2"/>
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="jordan12.html#T3">JORDAN12:3</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E14:136_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="txt" href="glib_001.html#E10:136_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E15:136_1_1_2"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">WB</font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">WB</font> <a href="glib_001.html#K4" target="_self">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, </i>;<br/><a NAME="E16:136_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">WB</font> <a href="glib_001.html#K4" target="_self">.last()</a> ,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><a NAME="E17:136_1_1_2"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><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">WB</font> is <a href="glib_001.html#V1" target="_self">closed</a> or  not <font color="Maroon">WB</font> is <a href="glib_001.html#V1" target="_self">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">WB</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 ;<br/><div class="add"><a NAME="E2:136_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:136_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K20" target="_self">.remove</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E4:136_1_1_2_2_1_1"/>
<font color="Maroon">W1</font> <a href="glib_001.html#K20" target="_self">.remove</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D1" target="_self">Def1</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K20" target="_self">.remove</a> 1,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, , <a class="txt" href="glib_001.html#E8:136_1_1_2"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span>

;<br/><a NAME="E6:136_1_1_2_2_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K20" target="_self">.remove</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 3
 ;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">W1</font> <a href="glib_001.html#K20" target="_self">.remove</a> 1,<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E8:136_1_1_2_2_1_1"/>
<font color="Maroon">W2</font> is <a href="glib_001.html#V2" target="_self">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">E91</font></i>: 
not <font color="Maroon">WB</font> is <a href="glib_001.html#V1" target="_self">closed</a>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">esb</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font>
 <b>by </b><i/>;<br/><a NAME="E4:136_1_1_2_2_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">esb</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><div><i><font color="Green">E94</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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">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">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a>, </i>;<br/><a NAME="E4:136_1_1_2_2_1_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E5:136_1_1_2_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E6:136_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2"/><i><font color="Green">E96</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">esb</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E8:136_1_1_2_2_1_2"/><i><font color="Green">E97</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E9:136_1_1_2_2_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E10:136_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
( <font color="Maroon">esb</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span> is <a href="funct_1.html#V2">one-to-one</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, , <a class="ref" href="funct_1.html#T48">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">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">WB</font> <a href="glib_001.html#K13" target="_self">.vertices()</a>  or  not <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">WB</font> <a href="glib_001.html#K13" target="_self">.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">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">WB</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">WB</font> &amp; <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">WB'</font> = <font color="Maroon">WB</font> as  <a href="glib_001.html#V2" target="_self">directed</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">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">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">WB</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="txt" href="glib_001.html#E10:136_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 <b>by </b><i/>;<br/><a NAME="E8:136_1_1_2_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i/>;<br/><a NAME="E9:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 ;<br/><a NAME="E10:136_1_1_2_2_1_2_2_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, </i>;<br/><b>then consider </b><font color="Maroon">naa1</font> being  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">naa1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i><a class="ref" href="glib_001.html#D5" target="_self">Def5</a></i>;<br/><b>set </b><font color="Maroon">es</font> = <font color="Maroon">esb</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">es</font> = <font color="Maroon">esb</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p4">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> as    <a href="graph_2.html#M1">FinSubsequence</a> of <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#T29">GRAPH_2:29</a></i>;<br/><a NAME="E14:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">es</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><a NAME="E15:136_1_1_2_2_1_2_2_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i>, , <a class="ref" href="tarski.html#D3">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">es</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">es</font> = <font color="Maroon">es</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/><a NAME="E18:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i/>;<br/><a NAME="E19:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E253</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E20:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">es</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T88">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">z</font> be    <a href="hidden.html#M1">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">E255</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> or  not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="xboole_0.html#D4">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">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">E256</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font> or <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font> )
 </span><b>by </b><i><a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="xboole_0.html#D2">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">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">esb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">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">E257</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T13">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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T25">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">E258</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">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">dom</a> <font color="Maroon">es</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="relat_1.html#T25">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">E259</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_2"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">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">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T99">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">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">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">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">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">E261</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><a NAME="E31:136_1_1_2_2_1_2_2_1_1"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">naa1</font>
 <b>by </b><i><a class="ref" href="pepin.html#T22">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">E263</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">naa1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><i><font color="Green">E264</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">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">naa1</font> <a href="nat_1.html#K5">div</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.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">naa1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="ref" href="xreal_1.html#T70">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">naa1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">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="ref" href="glib_001.html#D20" target="_self">Def20</a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</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">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="finseq_1.html#T7">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">E265</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>then reconsider </b><font color="Maroon">ses</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E38:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E266</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">ses</font> <a href="hidden.html#R1">=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E39:136_1_1_2_2_1_2_2_1_1"/>
<font color="Maroon">ses</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D19" target="_self">Def19</a>, <a class="ref" href="funct_1.html#T84">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">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="ref" href="finseq_4.html#T77">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">card</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i/>;<br/><a NAME="E42:136_1_1_2_2_1_2_2_1_1"/><b>then </b><i><font color="Green">E267</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, </i>;<br/><a NAME="E43:136_1_1_2_2_1_2_2_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y1</font>, <font color="Olive">y2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font> holds <br/><font color="Olive">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">y2</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">esbes1</font> = <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="funct_1.html#D1">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">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esbes1</font> )
 ;<br/><b>then consider </b><font color="Maroon">xa</font> being    <a href="hidden.html#M1">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">E271</font></i>: 
( <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ses</font> &amp; <font color="Maroon">ses</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">xa</font> = <font color="Maroon">xa</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a></i>;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xa</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green">E273</font></i>: 
( <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="relat_1.html#T86">RELAT_1:86</a></i>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green">E274</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">esbes1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="ref" href="funct_1.html#T8">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">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">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">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><b>then consider </b><font color="Maroon">xb</font> being    <a href="hidden.html#M1">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">E275</font></i>: 
( <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">xb</font> = <font color="Maroon">xb</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a></i>;<br/><a NAME="E13:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green">E276</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xa</font> &amp; <font color="Maroon">xa</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2 )
 <b>by </b><i><a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E14:136_1_1_2_2_1_2_2_1_1_5"/><i><font color="Green">E277</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xb</font> &amp; <font color="Maroon">xb</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="ref" href="finseq_1.html#T3">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">xb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">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">xb</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E17:136_1_1_2_2_1_2_2_1_1_5"/><b>then </b><i><font color="Green">E278</font></i>: 
<font color="Maroon">xb</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">xb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">xa</font>
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_1_5_1"/><b>then </b>
<font color="Maroon">xb</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="xxreal_0.html#T2">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">E279</font></i>: 
<font color="Maroon">xb</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="finseq_1.html#T3">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">[</a><span class="default"><font color="Maroon">xb</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="ref" href="funct_1.html#T8">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">[</a><span class="default"><font color="Maroon">xb</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ses</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="relat_1.html#D11">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">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ses</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">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">[</a><span class="default"><font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">esbes1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E46"><i><font color="Green">Lemma149</font></i></a>, <a class="ref" href="relat_1.html#D11">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#E46"><i><font color="Green">Lemma149</font></i></a>, <a class="ref" href="xboole_0.html#D4">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">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</font></i></a>, <a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="ref" href="finseq_1.html#D13">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">E280</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">esb</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">es</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>;<br/><a NAME="E47:136_1_1_2_2_1_2_2_1_1"/><i><font color="Green">E281</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="relat_1.html#T25">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">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green">E282</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.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">E283</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">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">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then </b><i><font color="Green">E284</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a>, <a class="ref" href="glib_001.html#D23" target="_self">Def23</a></i>
<br/>.= 
<font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><a NAME="E5:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</font></i></a>, <a class="ref" href="finseq_3.html#T27">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:136_1_1_2_2_1_2_2_1_1_6"/><b>then </b><i><font color="Green">E286</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="ref" href="finseq_1.html#D7">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">E287</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D15" target="_self">Def15</a>, <a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="funct_1.html#T21">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">E288</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E10:136_1_1_2_2_1_2_2_1_1_6"/><i><font color="Green">E289</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">naa1</font> <a href="nat_1.html#K5">div</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="glib_001.html#D22" target="_self">Def22</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">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="funct_1.html#T8">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">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">ses</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="relat_1.html#D11">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">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ses</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">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">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p5">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="ref" href="relat_1.html#D11">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">es</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="funct_1.html#T8">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">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E46"><i><font color="Green">Lemma149</font></i></a>, <a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="funct_1.html#T22">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">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">WB'</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<font color="Maroon">n</font> as   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, <a class="txt" href="glib_001.html#E12:136_1_1_2_2_1_2_2_1_1"><i><font color="Green">E104</font></i></a></i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E51:136_1_1_2_2_1_2_2_1_1"/>
<font color="Maroon">W2</font> is <a href="glib_001.html#V2" target="_self">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">E290</font></i>: 
not <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">WB</font> <a href="glib_001.html#K13" target="_self">.vertices()</a> 
 ;<br/><div class="add"><b>set </b><font color="Maroon">W2</font> = <font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_2"/>
( <font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> is <a href="glib_001.html#V2" target="_self">directed</a> &amp; <font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_001.html#D14" target="_self">Def14</a>, </i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">E291</font></i>: 
<font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#R1" target="_self">is_Walk_from</a> <font color="Maroon">W1</font> <a href="glib_001.html#K3" target="_self">.first()</a> ,<font color="Maroon">W1</font> <a href="glib_001.html#K4" target="_self">.last()</a> 
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">es</font> = <font color="Maroon">esb</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">E292</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">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">E293</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="funct_4.html#D1">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">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_1"/><i><font color="Green">E294</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="xboole_0.html#D2">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">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">E295</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_1_1_1_1"/><i><font color="Green">E296</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="finseq_1.html#T3">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">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</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">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="finseq_1.html#T3">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">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</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">E297</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</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">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="finseq_1.html#T3">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">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</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">E298</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">esb</font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">es</font> = <font color="Maroon">esb</font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D12">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_2"/><i><font color="Green">E299</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">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">E300</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">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">E301</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font> &amp; <font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="funct_1.html#T8">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">E303</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">E304</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, , <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="funct_4.html#D1">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">E319</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="tarski.html#D1">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">E320</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_2_2_1_1_1"/><i><font color="Green">E321</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</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">E322</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, </i>
<br/>.= 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="funct_1.html#T8">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">E323</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</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">E324</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, , <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="funct_4.html#D1">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">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, , <a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="xboole_0.html#D2">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">esb</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="funct_1.html#T8">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">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K8" target="_self">.cut</a> 1,<span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i>, <a class="ref" href="glib_001.html#D25" target="_self">Def25</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">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.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">es</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">es</font> = <font color="Maroon">es</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W1</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#D5">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">dom</a> <font color="Maroon">esb</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">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">E325</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">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">E326</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="xboole_0.html#D3">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">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">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">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="finseq_1.html#T3">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">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">E327</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> <a href="xboole_0.html#R1">misses</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E15:136_1_1_2_2_1_2_2_1_2"/><i><font color="Green">E328</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D8" target="_self">Def8</a></i>;<br/><b>then </b><i><font color="Green">E329</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">esb</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
;<br/><i><font color="Green">E330</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <font color="Maroon">es</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">esb</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="ref" href="pre_circ.html#T27">PRE_CIRC:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">esb</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T45">CQC_LANG:45</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#D26" target="_self">Def26</a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/><a NAME="E18:136_1_1_2_2_1_2_2_1_2"/><i><font color="Green">E331</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23" target="_self">Def23</a>, <a class="ref" href="xboole_1.html#T11">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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_6"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 ;<br/><a NAME="E2:136_1_1_2_2_1_2_2_1_2_6"/><b>then </b><i><font color="Green">E333</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_6"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, , <a class="ref" href="glib_001.html#D29" target="_self">Def29</a></i>;<br/><b>hence </b><a NAME="E4:136_1_1_2_2_1_2_2_1_2_6"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E334</font></i>:  <a href="finseq_1.html#K14">Sgm</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>
<b>by </b><i>, , <a class="ref" href="glib_001.html#D28" target="_self">Def28</a>, <a class="ref" href="finseq_3.html#T48">FINSEQ_3:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_3.html#T50">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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:136_1_1_2_2_1_2_2_1_2_8"/><i><font color="Green">E335</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.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">E336</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_8"/><i><font color="Green">E337</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D30" target="_self">Def30</a>, <a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="finseq_3.html#T27">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">Seq</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">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">E338</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E40"><i><font color="Green">Lemma142</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E6:136_1_1_2_2_1_2_2_1_2_8"/><i><font color="Green">E339</font></i>: 
 <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">esb</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">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">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span> or  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> ) )
 </span><b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="glib_001.html#D31" target="_self">Def31</a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E340</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E341</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">esb</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="ref" href="funct_1.html#T22">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">E342</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font> )
 <b>by </b><i>, <a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="ref" href="funct_1.html#T21">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">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="ref" href="finseq_1.html#D7">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">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="glib_001.html#D24" target="_self">Def24</a>, <a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</font></i></a>, <a class="ref" href="funct_4.html#T17">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">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">m</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/><i><font color="Green">E343</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font> )
 ;<br/><a NAME="E4:136_1_1_2_2_1_2_2_1_2_8_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8">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">E344</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E345</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="glib_001.html#D25" target="_self">Def25</a>, <a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <font color="Maroon">esb</font>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">esb</font></span>)</span>
<b>by </b><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">E346</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E41"><i><font color="Green">Lemma143</font></i></a>, <a class="ref" href="glib_001.html#D29" target="_self">Def29</a>, <a class="txt" href="glib_001.html#E42"><i><font color="Green">Lemma145</font></i></a>, <a class="txt" href="glib_001.html#E43"><i><font color="Green">Lemma146</font></i></a>, <a class="ref" href="finseq_1.html#T59">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">E347</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#D22" target="_self">Def22</a>, <a class="ref" href="tarski.html#D1">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">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">esb</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="glib_001.html#E45"><i><font color="Green">Lemma148</font></i></a>, <a class="txt" href="glib_001.html#E46"><i><font color="Green">Lemma149</font></i></a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">W1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">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">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E44"><i><font color="Green">Lemma147</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">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">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">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D27" target="_self">Def27</a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/><b>then reconsider </b><font color="Maroon">W2</font> = <font color="Maroon">WB</font> <a href="glib_001.html#K10" target="_self">.addEdge</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="funct_1.html#K1">.</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">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> as   <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font> <b>by </b><i><a class="ref" href="glib_001.html#D20" target="_self">Def20</a>, , , , <a class="ref" href="glib_001.html#D21" target="_self">Def21</a>, </i>;<br/><b>take </b><font color="Maroon">W2</font> = <font color="Maroon">W2</font>;<br/><b>thus </b><a NAME="E24:136_1_1_2_2_1_2_2_1_2"/>
<font color="Maroon">W2</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_001.html#D14" target="_self">Def14</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">W2</font> being  <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font> st <font color="Olive">W2</font> is <a href="glib_001.html#V2" target="_self">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">W2</font> being  <a href="glib_001.html#NM11">Path</a> of <font color="Maroon">W1</font> st <font color="Olive">W2</font> is <a href="glib_001.html#V2" target="_self">directed</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:136_1_1"/><b>then </b><i><font color="Green">E348</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:136_1_1"/><i><font color="Green">E349</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E6:136_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K15" target="_self">.length()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K15" target="_self">.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">Path</a> of <font color="Maroon">W</font> st <font color="Olive">b<sub>1</sub></font> is <a href="glib_001.html#V2" target="_self">directed</a>
 <b>by </b><i/>;<br/>


</div>
