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<div class="add">

<b>set </b><font color="Maroon">X</font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>;<br/>
<a NAME="E1:65_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f</font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">n</font>,<font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> <font color="Maroon">n</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">f</font> <a href="graphsp.html#R1" target="_self">hasBetterPathAt</a> <font color="Maroon">n</font>,<font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/>( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> implies ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> ) ) ) &amp; ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> implies ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> ) ) ) &amp; ( ( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) implies <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Olive">k</font> ) );<br/>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y2</font>] )
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">xx</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_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_1_1_1"/>
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) or ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1_1_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</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">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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">n</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_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_1_1_1_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_1_1_1_2"/><i><font color="Green">E63</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_1_1_1_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="graphsp.html#E1:65_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</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_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_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="E1:65_1_1_1_1_2_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] or  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_2_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">y1</font> = 
 <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_2_1_1_2"/><i><font color="Green">E66</font></i>: 
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>hence </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, </i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, , <a class="ref" href="graphsp.html#T5" target="_self">Th5</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_1_1_1_3"><b>suppose </b></a><a NAME="E1:65_1_1_1_1_3"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>hence </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<font color="Maroon">y2</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:65_1_1_1"><i><font color="Green">E53</font></i></a>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">xx</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:65_1_1_2"/>
<font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/><b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">xx</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:65_1_1_2_1"/>
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) or ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:65_1_1_2_1_1"/>
 ex <font color="Olive">y1</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Olive">y1</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_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="E1:65_1_1_2_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_1_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">y1</font> = <font color="Maroon">f</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">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E2:65_1_1_2_1_1_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>]
 <b>by </b><i>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_1_1_1_2"/><i><font color="Green">E71</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">y1</font> = <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E2:65_1_1_2_1_1_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>]
 <b>by </b><i>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<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_1_2_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_2"/><i><font color="Green">E84</font></i>: 
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:65_1_1_2_1_2"/>
 ex <font color="Olive">y1</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Olive">y1</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_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="E1:65_1_1_2_1_2_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] or  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_2_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">y1</font> =  <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/><b>thus </b><a NAME="E2:65_1_1_2_1_2_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>]
 <b>by </b><i>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_2_1_1_2"/><i><font color="Green">E98</font></i>: 
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>take </b><font color="Maroon">y1</font> = <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E2:65_1_1_2_1_2_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>]
 <b>by </b><i>, <a class="ref" href="graphsp.html#T5" target="_self">Th5</a></i>;<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_1_2_1_3"><b>suppose </b></a><a NAME="E1:65_1_1_2_1_3"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:65_1_1_2_1_3"/>
 ex <font color="Olive">y1</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Olive">y1</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">y1</font> = <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>;<br/>

<b>thus </b><a NAME="E1:65_1_1_2_1_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">xx</font>,<font color="Maroon">y1</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:65_1_1"/><i><font color="Green">E100</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="graphsp.html#E2:65_1_1"><i><font color="Green">E52</font></i></a>, );<br/></i>
<a NAME="E6:65_1_1"/><i><font color="Green">E101</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y1</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:65_1_1_3"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xx</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:65_1_1_3"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">xx</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xx</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">xx</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_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_1_3_1"/>
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) or ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) or <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_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_1_3_1_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] or  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><a NAME="E2:65_1_1_3_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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">n</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#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma57</font></i></a></i>;<br/><b>hence </b><a NAME="E3:65_1_1_3_1_1_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_1_1_1_2"/>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>then </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma57</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:65_1_1_3_1_1_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<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_1_3_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_2"/><i><font color="Green">E104</font></i>: 
( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_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_1_3_1_2_1_1"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] or  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_2_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><a NAME="E2:65_1_1_3_1_2_1_1_1"/><b>then </b>
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma57</font></i></a></i>;<br/><b>hence </b><a NAME="E3:65_1_1_3_1_2_1_1_1"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_1_1_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_2_1_1_2"/>
 not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div class="add"><b>then </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="txt" href="graphsp.html#E7"><i><font color="Green">Lemma57</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:65_1_1_3_1_2_1_1_2"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<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_1_3_1_3"><b>suppose </b></a><a NAME="E1:65_1_1_3_1_3"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>then </b><font color="Maroon">y1</font> = 
<font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="graphsp.html#E6"><i><font color="Green">Lemma56</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E3:65_1_1_3_1_3"/>
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:65_1_1"/>
<font color="Maroon">F</font> is   <a href="finseq_1.html#NM1">FinSequence</a>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E8:65_1_1"/><b>then </b>
<font color="Maroon">F</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="graphsp.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as    <a href="finseq_2.html#M2">Element</a> of <a href="numbers.html#K1">REAL</a>  <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E10:65_1_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E11:65_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>hence </b><a NAME="E12:65_1_1"/>
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> implies ( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] implies <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</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">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</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">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>] implies <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> ) ) ) &amp; ( 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> implies ( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] implies <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="graphsp.html#K18" target="_self">newpathcost</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font>] implies <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> ) ) ) &amp; ( ( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font> ) implies <font color="Maroon">F</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="goboard1.html#K1">.</a> <font color="Maroon">k</font> ) )
 <b>by </b><i/>;<br/>


</div>
