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<b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:63_1_2"/><i><font color="Green" title="E50">A24</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/>( ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> implies ( ( <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c3">F1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( not <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c3">F1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> ) ) ) &amp; ( 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> implies ( ( <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> implies <font color="Maroon" title="c3">F1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> ) &amp; ( not <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> implies <font color="Maroon" title="c3">F1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> ) ) ) &amp; ( ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> ) implies <font color="Maroon" title="c3">F1</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> ) ) ) )
 <b>and </b><br/><a NAME="E2:63_1_2"/><i><font color="Green" title="E51">A25</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font> holds <br/>( ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> implies ( ( <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c4">F2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( not <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c4">F2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> ) ) ) &amp; ( 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> implies ( ( <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> implies <font color="Maroon" title="c4">F2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> ) &amp; ( not <font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> implies <font color="Maroon" title="c4">F2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> ) ) ) &amp; ( ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> ) implies <font color="Maroon" title="c4">F2</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Olive" title="b1">k</font> ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">xx</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:63_1_2_1"/><i><font color="Green" title="E52">A26</font></i>: 
<font color="Maroon" title="c5">xx</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">F1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">xx</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[] <b>means </b><font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c2">n</font>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[] <b>means </b><font color="Maroon" title="c1">f</font> <a href="graphsp.html#R1" title="GRAPHSP:pred.1">hasBetterPathAt</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c6">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_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="E1:63_1_2_1_1"/>
( ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> ) or ( 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> ) or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_1"/><i><font color="Green" title="E53">A27</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_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="E1:63_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="63_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_1_1_1_1"/><i><font color="Green" title="E54">A28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> = 
<font color="Maroon" title="c1">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default">3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_1"><i><font color="Green" title="E53">A27</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E2:63_1_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_1"><i><font color="Green" title="E53">A27</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_1_1_1_1"><i><font color="Green" title="E54">A28</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_1_1_1_2"/><i><font color="Green" title="E54">A29</font></i>: 
 not <font color="Maroon">S<sub>1</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> = 
<font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_1"><i><font color="Green" title="E53">A27</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E2:63_1_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_1"><i><font color="Green" title="E53">A27</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_1_1_1_2"><i><font color="Green" title="E54">A29</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="63_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_2"/><i><font color="Green" title="E53">A30</font></i>: 
( 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><div class="add"><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_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="E1:63_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="63_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_2_1_1_1"/><i><font color="Green" title="E54">A31</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> = 
 <a href="graphsp.html#K17" title="GRAPHSP:func.17">newpathcost</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_2"><i><font color="Green" title="E53">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E2:63_1_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_2"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_2_1_1_1"><i><font color="Green" title="E54">A31</font></i></a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_2_1_1_2"/><i><font color="Green" title="E54">A32</font></i>: 
 not <font color="Maroon">S<sub>2</sub></font>[]
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> = 
<font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_2"><i><font color="Green" title="E53">A30</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E2:63_1_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_2"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_2_1_1_2"><i><font color="Green" title="E54">A32</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="63_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:63_1_2_1_1_3"/><i><font color="Green" title="E53">A33</font></i>: 
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 3 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c3">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font> = 
<font color="Maroon" title="c1">f</font> <a href="goboard1.html#K1" title="GOBOARD1:func.1">.</a> <font color="Maroon" title="c6">k</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xx</font>
<b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E2:63_1_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1"><i><font color="Green" title="E52">A26</font></i></a>, <a class="txt" href="graphsp.html#E1:63_1_2_1_1_3"><i><font color="Green" title="E53">A33</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="E4:63_1_2"/>
<font color="Maroon" title="c3">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F2</font>
 <b>by </b><i><a class="txt" href="graphsp.html#E1:63_1_2"><i><font color="Green" title="E50">A24</font></i></a>, <a class="txt" href="graphsp.html#E2:63_1_2"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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