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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:65_1_2"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> implies ( ( <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( not <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) &amp; ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> implies ( ( <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ) &amp; ( not <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) &amp; ( ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> ) implies <font color="Maroon">c<sub>3</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 <b>and </b><br/><a NAME="E2:65_1_2"/><i><font color="Green">E52</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> implies ( ( <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; ( not <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) &amp; ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> implies ( ( <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> ) &amp; ( not <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) &amp; ( ( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> ) implies <font color="Maroon">c<sub>4</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[] means <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[] means <font color="Maroon">c<sub>1</sub></font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_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="E1:65_1_2_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> ) or ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> ) or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_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="E1:65_1_2_1_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[] or  not <font color="Maroon">S<sub>1</sub></font>[] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E2:65_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_1_1_1_1"><i><font color="Green">E55</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E2:65_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_1_1_1_2"><i><font color="Green">E55</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_2"/><i><font color="Green">E54</font></i>: 
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_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="E1:65_1_2_1_1_2_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[] or  not <font color="Maroon">S<sub>2</sub></font>[] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="binarith.html#K5">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_2"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E2:65_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
 not <font color="Maroon">S<sub>2</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_2"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E2:65_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_2_1_1_2"><i><font color="Green">E55</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:65_1_2_1_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="goboard1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E2:65_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_2_1_1_3"><i><font color="Green">E54</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="graphsp.html#E2:65_1_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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