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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V7">real-weighted</a> <a href="glib_003.html#V9">real-elabeled</a> <a href="glib_005.html#V1" target="_self">complete-elabeled</a> <a href="glib_003.html#NM4">WEGraph</a>;<br/><b>let </b><font color="Maroon">source</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">sink</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P</font> be  <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">source</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sink</font> &amp; <font color="Maroon">G</font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> )
 ;<br/>

<b>set </b><font color="Maroon">G2</font> =  <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font>;<br/>
<a NAME="E2:77"/><i><font color="Green">E51</font></i>: 
(  <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font> &amp;  <a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K5">the_Weight_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">EL1</font> =  <a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">EL2</font> =  <a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span>;<br/>
<a NAME="E3:77"/><i><font color="Green">E82</font></i>: 
 <a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><a NAME="E1:77_1"/>
( <font color="Maroon">source</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">sink</font> is   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E2:77_1"/>
( <font color="Maroon">source</font> is   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> &amp; <font color="Maroon">sink</font> is   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span>
 ;<br/><a NAME="E2:77_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E3:77_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_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="E1:77_1_1_1_1"/>
(  not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  or <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_1_1_1_1"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:77_1_1_1_1_2"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T101">GLIB_001:101</a></i>;<br/><a NAME="E4:77_1_1_1_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E5:77_1_1_1_1_2"/>
( <font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><a NAME="E6:77_1_1_1_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><a NAME="E7:77_1_1_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> 
 <b>by </b><i/>;<br/><div><i><font color="Green">E109</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_1_1_2_1"/>
( <font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> &amp; <font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font> )
 ;<br/><a NAME="E2:77_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E3:77_1_1_1_1_2_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:77_1_1_1_1_2_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><a NAME="E5:77_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/><a NAME="E6:77_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">sink</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><b>hence </b><a NAME="E7:77_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">PFS</font> = <font color="Maroon">P</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> ;<br/><b>set </b><font color="Maroon">n1div2</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2;<br/><a NAME="E9:77_1_1_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:77_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_001.html#T78">GLIB_001:78</a></i>;<br/><a NAME="E11:77_1_1_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E12:77_1_1_1_1_2"/>
2 <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T22">PEPIN:22</a></i>;<br/><a NAME="E13:77_1_1_1_1_2"/><b>then </b><i><font color="Green">E128</font></i>: 
2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/><a NAME="E14:77_1_1_1_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E15:77_1_1_1_1_2"/><i><font color="Green">E130</font></i>: 
<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">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="glib_005.html#D3" target="_self">Def3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_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="E1:77_1_1_1_1_2_2_1"/>
( <font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> or <font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font> )
 </span><b>by </b><i>, <a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:77_1_1_1_1_2_2_1_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:77_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green">E132</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E3:77_1_1_1_1_2_2_1_1"/>
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E4:77_1_1_1_1_2_2_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E5:77_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, </i>;<br/><a NAME="E6:77_1_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:77_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E8:77_1_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p4">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E9:77_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:77_1_1_1_1_2_2_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:77_1_1_1_1_2_2_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E3:77_1_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , </i>;<br/><a NAME="E4:77_1_1_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:77_1_1_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:77_1_1_1_1_2_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E7:77_1_1_1_1_2_2_1_2"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/><a NAME="E8:77_1_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font></span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i>, <a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E9:77_1_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:77_1_1_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:77_1"/>
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> holds <br/>( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5">the_Weight_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Olive">e</font> )
 ;<br/><b>let </b><font color="Maroon">v</font> be   <a href="glib_000.html#NM10">Vertex</a> of <span class="p1">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span>;<br/><b>assume </b><a NAME="E5:77_1"/><i><font color="Green">E135</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">source</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">sink</font> )
 ;<br/><b>reconsider </b><font color="Maroon">v'</font> = <font color="Maroon">v</font> as   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font> <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_000.html#D36">GLIB_000:def 36</a></i>;<br/><a NAME="E7:77_1"/><i><font color="Green">E137</font></i>: 
( <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  &amp; <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_000.html#T99">GLIB_000:99</a></i>;<br/><a NAME="E8:77_1"/><i><font color="Green">E138</font></i>: 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.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="E3:77_1_2_1_1"/><i><font color="Green">E139</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88">GLIB_001:88</a></i>;<br/><div><i><font color="Green">E140</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/><b>then </b><font color="Maroon">v</font> = 
<font color="Maroon">P</font> <a href="glib_001.html#K4">.last()</a> 
<b>by </b><i>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon">sink</font>
<b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
;<br/><b>hence </b><a NAME="E3:77_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E5:77_1_2_1_1"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E6:77_1_2_1_1"/><i><font color="Green">E142</font></i>: 
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/><div><i><font color="Green">E143</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><b>then </b><font color="Maroon">v</font> = 
<font color="Maroon">P</font> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i>, <a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>
<br/>.= 
<font color="Maroon">source</font>
<b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>
;<br/><b>hence </b><a NAME="E3:77_1_2_1_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><a NAME="E8:77_1_2_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E9:77_1_2_1_1"/><b>then </b>
1 <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="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then reconsider </b><font color="Maroon">n2</font> = <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> as  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><b>set </b><font color="Maroon">e1</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">e2</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">T</font> = <font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> ;<br/><a NAME="E11:77_1_2_1_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</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/><a NAME="E12:77_1_2_1_1"/><b>then </b><i><font color="Green">E145</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E13:77_1_2_1_1"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D3">GLIB_001:def 3</a></i>;<br/><a NAME="E14:77_1_2_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><div><i><font color="Green">E148</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_3"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><a NAME="E2:77_1_2_1_1_3"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E3:77_1_2_1_1_3"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E150</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_4"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_4"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:77_1_2_1_1_4"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E4:77_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_5"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">G</font> )
 ;<br/><b>then </b><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>
;<br/><b>hence </b><a NAME="E3:77_1_2_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E155</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_6"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 ;<br/><b>then </b><i><font color="Green">E157</font></i>: <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>
;<br/><a NAME="E3:77_1_2_1_1_6"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:77_1_2_1_1_6"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E5:77_1_2_1_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><a NAME="E19:77_1_2_1_1"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E20:77_1_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E21:77_1_2_1_1"/><b>then </b><i><font color="Green">E159</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T139">GLIB_001:139</a></i>;<br/><div><i><font color="Green">E160</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_7"/><i><font color="Green">E161</font></i>: 
( ( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  or <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  ) &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_7_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">m</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="E3:77_1_2_1_1_7_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/><a NAME="E4:77_1_2_1_1_7_1"/><i><font color="Green">E163</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</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/><div><i><font color="Green">E164</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_7_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_7_1_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  or <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T59">GLIB_000:59</a></i>;<br/><b>hence </b><a NAME="E3:77_1_2_1_1_7_1_1_1_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_7_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T61">GLIB_000:61</a></i>;<br/><b>hence </b><a NAME="E3:77_1_2_1_1_7_1_1_1_2"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_7_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_7_1_2_1"/>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> or <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_1"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_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="E1:77_1_2_1_1_7_1_2_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_7_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_7_1_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_1_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_7_1_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_7_1_2_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_2"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_7_1_2_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_7_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_7_1_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_7_1_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_7_1_2_1_2_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_7_1_2_1_2_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_7_1_2_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:77_1_2_1_1_7_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_7"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1"/>
( ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> ) or ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font> ) or ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> ) or ( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font> ) )
 </span><b>by </b><i>, <a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1"/><i><font color="Green">E167</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">XIN</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">XOUT</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:77_1_2_1_1_8_1_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T60">GLIB_000:60</a>, <a class="ref" href="glib_000.html#T62">GLIB_000:62</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_1"/><i><font color="Green">E169</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><i><font color="Green">E170</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XIN</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:77_1_2_1_1_8_1_1"/><i><font color="Green">E171</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><i><font color="Green">E172</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XOUT</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:77_1_2_1_1_8_1_1"/><i><font color="Green">E173</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_1_3"/>
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_1_3"/><b>then </b><i><font color="Green">E174</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_1_3"/><b>then </b><i><font color="Green">E175</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T59">GLIB_000:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_1"/><i><font color="Green">E176</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , , </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2"/><b>then </b>
not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E178</font></i>: <font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">m</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="E3:77_1_2_1_1_8_1_1_3_1_1_2_2"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/><a NAME="E4:77_1_2_1_1_8_1_1_3_1_1_2_2"/><i><font color="Green">E180</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_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="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 </span><b>by </b><i><a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_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="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E182</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, , , <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/><i><font color="Green">E183</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E4:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, <a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:77_1_2_1_1_8_1_1_3_1_1_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><a NAME="E11:77_1_2_1_1_8_1_1"/><b>then </b><i><font color="Green">E185</font></i>: 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <font color="Maroon">XIN</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E12:77_1_2_1_1_8_1_1"/><i><font color="Green">E186</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green">E187</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_1_4"/>
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_1_4"/><b>then </b><i><font color="Green">E188</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_1_4"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T61">GLIB_000:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_1"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , , , <a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2"/><b>then </b>
not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E192</font></i>: <font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon">v1</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> being   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">m</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="E3:77_1_2_1_1_8_1_1_4_1_1_2_2"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> &amp; <font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104">GLIB_001:104</a></i>;<br/><a NAME="E4:77_1_2_1_1_8_1_1_4_1_1_2_2"/><i><font color="Green">E194</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0
 <b>by </b><i><a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_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="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 </span><b>by </b><i><a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_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="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_005.html#D12" target="_self">Def12</a>, , <a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> or <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> or <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E197</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, , , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E4:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_3"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><div class="add"><b>hence </b><a NAME="E2:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="glib_005.html#D12" target="_self">Def12</a>, <a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, , <a class="ref" href="glib_001.html#D28">GLIB_001:def 28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:77_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:77_1_2_1_1_8_1_1_4_1_1_2_2"/>
contradiction
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div> <a href="uproots.html#K3">Sum</a> <font color="Maroon">XIN</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">XOUT</font>
<b>by </b><i><a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
;<br/><b>hence </b><a NAME="E15:77_1_2_1_1_8_1_1"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_2"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_2"/><b>then </b><i><font color="Green">E199</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , </i>;<br/><a NAME="E3:77_1_2_1_1_8_1_2"/><i><font color="Green">E200</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T60">GLIB_000:60</a></i>;<br/><b>set </b><font color="Maroon">XIN1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">XIN2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><i><font color="Green">E201</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></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">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XIN1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><i><font color="Green">E202</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">XIN1</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XIN2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:77_1_2_1_1_8_1_2"/><i><font color="Green">E203</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:77_1_2_1_1_8_1_2"/><i><font color="Green">E204</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><div><i><font color="Green">E205</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_2_3"/><i><font color="Green">E206</font></i>: 
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_2_3"/><b>then </b><i><font color="Green">E207</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_2_3"/><i><font color="Green">E208</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_2_3_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or ( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_2_3_1_1_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_2_3_1_1_1"/><b>then </b><i><font color="Green">E211</font></i>: 
( not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<font color="Maroon">XIN1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E4:77_1_2_1_1_8_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_2_3_1_1_2"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E4:77_1_2_1_1_8_1_2_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_3_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_2_3_1_1_3"/><i><font color="Green">E213</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_2_3_1_1_3"/><b>then </b><i><font color="Green">E214</font></i>: 
( not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E215</font></i>: <font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<font color="Maroon">XIN1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:77_1_2_1_1_8_1_2_3_1_1_3"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, , , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_2_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><a NAME="E11:77_1_2_1_1_8_1_2"/>
not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E216</font></i>: <font color="Maroon">XIN1</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E13:77_1_2_1_1_8_1_2"/><i><font color="Green">E217</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a>  &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_2_5"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_2_5"/><b>then </b><i><font color="Green">E219</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_2_5"/><i><font color="Green">E220</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, , </i>;<br/><a NAME="E4:77_1_2_1_1_8_1_2_5"/><i><font color="Green">E221</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="glib_000.html#T61">GLIB_000:61</a></i>;<br/><a NAME="E5:77_1_2_1_1_8_1_2_5"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a>, <a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E6:77_1_2_1_1_8_1_2_5"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a>, </i>;<br/><a NAME="E7:77_1_2_1_1_8_1_2_5"/><b>then </b>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, , <a class="ref" href="glib_005.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E8:77_1_2_1_1_8_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, , , , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><a NAME="E15:77_1_2_1_1_8_1_2"/><b>then </b><i><font color="Green">E223</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/> <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">XIN2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">XIN1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">XIN1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p0">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E17:77_1_2_1_1_8_1_2"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_3"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_3"/><b>then </b><i><font color="Green">E225</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , </i>;<br/><a NAME="E3:77_1_2_1_1_8_1_3"/><i><font color="Green">E226</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T62">GLIB_000:62</a></i>;<br/><b>set </b><font color="Maroon">XOUT1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">XOUT2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><i><font color="Green">E227</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></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">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XOUT1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><i><font color="Green">E228</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">XOUT1</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XOUT2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:77_1_2_1_1_8_1_3"/><i><font color="Green">E231</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:77_1_2_1_1_8_1_3"/><i><font color="Green">E232</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><div><i><font color="Green">E233</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_3_3"/><i><font color="Green">E234</font></i>: 
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_3_3"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_3_3"/><i><font color="Green">E237</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_3_3_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or ( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_3_3_1_1_1"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_3_3_1_1_1"/><b>then </b><i><font color="Green">E240</font></i>: 
( not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<font color="Maroon">XOUT1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E4:77_1_2_1_1_8_1_3_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_3_3_1_1_2"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_3_3_1_1_2"/><b>then </b>
<font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E4:77_1_2_1_1_8_1_3_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_3_1_1_3"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_3_3_1_1_3"/><i><font color="Green">E244</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_3_3_1_1_3"/><b>then </b><i><font color="Green">E245</font></i>: 
( not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E246</font></i>: <font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<font color="Maroon">XOUT1</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:77_1_2_1_1_8_1_3_3_1_1_3"/>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, , , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_3_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT2</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 ;<br/></div><b>end;</b></div><a NAME="E11:77_1_2_1_1_8_1_3"/>
not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E247</font></i>: <font color="Maroon">XOUT1</font> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><i><font color="Green">E248</font></i>:  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">XOUT2</font>
<b>by </b><i>, , , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">XOUT1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <font color="Maroon">XOUT1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p0">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>

;<br/><a NAME="E14:77_1_2_1_1_8_1_3"/><i><font color="Green">E249</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a>  &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a>  )
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_3_6"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_3_6"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_3_6"/><i><font color="Green">E252</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, , <a class="ref" href="glib_005.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E4:77_1_2_1_1_8_1_3_6"/><i><font color="Green">E253</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, , <a class="ref" href="glib_000.html#T59">GLIB_000:59</a></i>;<br/><a NAME="E5:77_1_2_1_1_8_1_3_6"/><b>then </b><i><font color="Green">E254</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a>, <a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E6:77_1_2_1_1_8_1_3_6"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D11" target="_self">Def11</a>, , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a>, </i>;<br/><a NAME="E7:77_1_2_1_1_8_1_3_6"/><b>then </b>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i><a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, , , </i>;<br/><b>hence </b><a NAME="E8:77_1_2_1_1_8_1_3_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D10" target="_self">Def10</a>, , <a class="ref" href="glib_005.html#D9" target="_self">Def9</a>, , </i>;<br/></div><b>end;</b></div><a NAME="E16:77_1_2_1_1_8_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E17:77_1_2_1_1_8_1_3"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_004.html#T7">GLIB_004:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_4"/><i><font color="Green">E255</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n2</font>,<font color="Maroon">G</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_4"/><b>then </b><i><font color="Green">E256</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , , </i>;<br/><a NAME="E3:77_1_2_1_1_8_1_4"/><i><font color="Green">E257</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_000.html#T60">GLIB_000:60</a>, <a class="ref" href="glib_000.html#T62">GLIB_000:62</a></i>;<br/><b>set </b><font color="Maroon">XIN</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">XOUT</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:77_1_2_1_1_8_1_4"/><i><font color="Green">E258</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="cqc_lang.html#T5">CQC_LANG:5</a></i>;<br/><b>then </b><i><font color="Green">E259</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XIN</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><i><font color="Green">E260</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p0">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon">XOUT</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1">Rbag</a> of <font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:77_1_2_1_1_8_1_4"/><i><font color="Green">E261</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a>  &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_4_3"/>
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_4_3"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_4_3"/><b>then </b><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T59">GLIB_000:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_4_3_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_3_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_4_3_1_1_1"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_4_3_1_1_1"/><b>then </b>
<font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_3_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_4_3_1_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_4_3_1_1_2"/><b>then </b>
not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E266</font></i>: <font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:77_1_2_1_1_8_1_4_3_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E5:77_1_2_1_1_8_1_4_3_1_1_2"/><b>then </b>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E6:77_1_2_1_1_8_1_4_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_4_3"/>
<font color="Maroon">XIN</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><b>then </b><i><font color="Green">E267</font></i>:  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">XIN</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_1_8_1_4_5"/>
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_1_8_1_4_5"/><b>then </b><i><font color="Green">E268</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_1_8_1_4_5"/><b>then </b><i><font color="Green">E269</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T61">GLIB_000:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:77_1_2_1_1_8_1_4_5_1_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_5_1_1_1"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_4_5_1_1_1"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_4_5_1_1_1"/><b>then </b>
<font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, , <a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_1_8_1_4_5_1_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_1_8_1_4_5_1_1_2"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_1_8_1_4_5_1_1_2"/><b>then </b>
not <font color="Maroon">e</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">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green">E272</font></i>: <font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:77_1_2_1_1_8_1_4_5_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_000.html#D16">GLIB_000:def 16</a></i>;<br/><a NAME="E5:77_1_2_1_1_8_1_4_5_1_1_2"/><b>then </b>
not <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 <b>by </b><i>, <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, , </i>;<br/><b>hence </b><a NAME="E6:77_1_2_1_1_8_1_4_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D8" target="_self">Def8</a>, , , <a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_1_8_1_4_5"/>
<font color="Maroon">XOUT</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D7" target="_self">Def7</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><b>then </b> <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3">Sum</a> <font color="Maroon">XOUT</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p4">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p5">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K9" target="_self">.tolerance()</a> </span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a></i>
;<br/><b>hence </b><a NAME="E14:77_1_2_1_1_8_1_4"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E24:77_1_2_1_1"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="77_1_2_1_2"><b>suppose </b></a><a NAME="E1:77_1_2_1_2"/><i><font color="Green">E273</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K13">.vertices()</a> 
 ;<br/><div class="add"><a NAME="E2:77_1_2_1_2"/><i><font color="Green">E274</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> 
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_2"/><b>then </b><i><font color="Green">E275</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_2_1"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_2_1"/><b>then </b><i><font color="Green">E277</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_2_1_1"/><i><font color="Green">E278</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:77_1_2_1_2_1_1"/><i><font color="Green">E279</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">x</font>,<font color="Maroon">v</font>, <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#T60">GLIB_000:60</a></i>;<br/><a NAME="E4:77_1_2_1_2_1_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">v</font>, <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><a NAME="E5:77_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">x</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:77"><i><font color="Green">E37</font></i></a>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><b>hence </b><a NAME="E6:77_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_001.html#T106">GLIB_001:106</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:77_1_2_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , , , , </i>;<br/></div><b>end;</b></div><a NAME="E5:77_1_2_1_2"/><b>then </b><i><font color="Green">E280</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E6:77_1_2_1_2"/><i><font color="Green">E281</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a>  &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a>  )
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:77_1_2_1_2_2"/><i><font color="Green">E282</font></i>: 
<font color="Maroon">e</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"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E2:77_1_2_1_2_2"/><b>then </b><i><font color="Green">E283</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> 
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:77_1_2_1_2_2"/><i><font color="Green">E284</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_2_1_2_2_1"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="glib_001.html#K14">.edges()</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:77_1_2_1_2_2_1"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">v</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, , <a class="ref" href="glib_000.html#T62">GLIB_000:62</a></i>;<br/><a NAME="E4:77_1_2_1_2_2_1"/>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">v</font>,<font color="Maroon">x</font>,<font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:77"><i><font color="Green">E51</font></i></a>, <a class="ref" href="glib_000.html#T19">GLIB_000:19</a></i>;<br/><b>hence </b><a NAME="E5:77_1_2_1_2_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, <a class="ref" href="glib_001.html#T106">GLIB_001:106</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:77_1_2_1_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , , </i>;<br/></div><b>end;</b></div><a NAME="E8:77_1_2_1_2"/><b>then </b><i><font color="Green">E287</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><b>thus </b> <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v'</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="glib_004.html#T7">GLIB_004:7</a></i>
<br/>.= 
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D1" target="_self">Def1</a>, <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_004.html#T7">GLIB_004:7</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E10:77_1"/>
 <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K1" target="_self">the_ELabel_of</a> <span class="p3">(<span class="default"><a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span></span>)</span> <a href="pre_circ.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="glib_000.html#K28">.edgesOut()</a> </span>)</span></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:77"/>
 <a href="glib_005.html#K12" target="_self">FF:AugmentPath</a> <font color="Maroon">G</font>,<font color="Maroon">P</font> <a href="glib_005.html#R1" target="_self">has_valid_flow_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font>
 <b>by </b><i/>;<br/>


</div>
