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

<b>set </b><font color="Maroon">F</font> =  <a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:57_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:57_1_1_1"/><i><font color="Green">E37</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"><a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> <a href="seq_1.html#K1">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:57_1_1_1"/><i><font color="Green">E51</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="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_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="E1:57_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>  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="57_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_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"><a NAME="E2:57_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:57_1_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_1_1_1_2"/><i><font color="Green">E82</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/><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="E3:57_1_1_1_1_1_2"/><i><font color="Green">E89</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="E4:57_1_1_1_1_1_2"/><i><font color="Green">E92</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#T137">GLIB_001:137</a>, </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> or  not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:57_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</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="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="glib_005.html#K10" target="_self">.tolerance()</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:57_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E93</font></i>: 
not <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="glib_000.html#R2">DJoins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2</span>)</span>,<font color="Maroon">G</font>
 ;<br/><div class="add"><a NAME="E2:57_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</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#K10" target="_self">.tolerance()</a> </span>)</span>
 <b>by </b><i>, , </i>;<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="E3:57_1_1_1_1_1_2_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="nat_1.html#T29">NAT_1:29</a>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:57_1_1_1_1_1_2_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">e</font> <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="E5:57_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E95</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="E6:57_1_1_1_1_1_2_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="E7:57_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E96</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="E8:57_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E109</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="E9:57_1_1_1_1_1_2_1_1_2"/>
<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> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:57_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E10:57_1_1_1_1_1_2_1_1_2"/><b>then </b>
<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_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E11:57_1_1_1_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:57_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">P</font> <a href="glib_005.html#K10" target="_self">.tolerance()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K6">the_ELabel_of</a> <font color="Maroon">G</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:57_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <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/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:57_1_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:57_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E2:57_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:57_1_1"/>
 <a href="glib_005.html#K11" target="_self">FF:PushFlow</a> <font color="Maroon">G</font>,<font color="Maroon">P</font> is <a href="seqm_3.html#V6">natural-yielding</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D8">SEQM_3:def 8</a></i>;<br/>


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