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

<b>let </b><font color="Maroon" title="c1">W</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">qe</font>, <font color="Maroon" title="c4">pe</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>







<b>set </b><font color="Maroon" title="c5">D</font> = the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>
<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E61">A1</font></i>: 
( <font color="Maroon" title="c3">qe</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">qe</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">pe</font> &amp; <font color="Maroon" title="c1">W</font> <a href="graph_5.html#R5" title="GRAPH_5:pred.5">is_weight&gt;=0of</a> <font color="Maroon" title="c2">G</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b2">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">W</font>;<br/>
<a NAME="E2:77"/><i><font color="Green" title="E62">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/>

<b>assume </b><a NAME="E1:77_1"/>
( <font color="Maroon" title="c7">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/>

<a NAME="E2:77_1"/><b>then </b>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E3:77_1"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T55" target="_self" title="GRAPH_5:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E4:77_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="graph_5.html#T54" target="_self" title="GRAPH_5:th.54">Th54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:77"/><i><font color="Green" title="E63">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:77_2"/><i><font color="Green" title="E64">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>;<br/><b>assume </b><a NAME="E1:77_2_1"/><i><font color="Green" title="E65">A5</font></i>: 
( <font color="Maroon" title="c8">q</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><b>then consider </b><font color="Maroon" title="c9">q1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:77_2_1"/><i><font color="Green" title="E66">A6</font></i>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T21" title="FINSEQ_2:th.21">FINSEQ_2:21</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">p1</font>, <font color="Maroon" title="c12">p2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E5:77_2_1"/><i><font color="Green" title="E67">A7</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c12">p2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c12">p2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77_2_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="graph_5.html#E3:77_2_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="graph_5.html#T12" target="_self" title="GRAPH_5:th.12">Th12</a></i>;<br/><a NAME="E6:77_2_1"/><i><font color="Green" title="E68">A8</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">q1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77_2_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="graph_5.html#E3:77_2_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>;<br/><a NAME="E7:77_2_1"/><i><font color="Green" title="E69">A9</font></i>: 
<font color="Maroon" title="c11">p1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E5:77_2_1"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">p1</font> = <font color="Maroon" title="c11">p1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">y</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E7:77_2_1"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">p2</font> = <font color="Maroon" title="c12">p2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E5:77_2_1"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">q1</font> = <font color="Maroon" title="c9">q1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <font color="Maroon" title="c2">G</font> <b>by </b><i><a class="txt" href="graph_5.html#E3:77_2_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/><a NAME="E12:77_2_1"/><i><font color="Green" title="E70">A10</font></i>: 
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c1">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c16">q1</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c1">W</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="graph_5.html#E3:77_2_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></i>;<br/><i><font color="Green" title="E71">A11</font></i>:  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">W</font> = 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">y</font></span>)</span>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c1">W</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="graph_5.html#E5:77_2_1"><i><font color="Green" title="E67">A7</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c1">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c1">W</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c13">p1</font>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c15">p2</font>,<font color="Maroon" title="c1">W</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c1">W</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c15">p2</font></span>)</span>,<font color="Maroon" title="c1">W</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c14">y</font>,<font color="Maroon" title="c1">W</font></span>)</span>
<b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="graph_5.html#T50" target="_self" title="GRAPH_5:th.50">Th50</a>, <a class="ref" href="graph_5.html#T58" target="_self" title="GRAPH_5:th.58">Th58</a></i>
;<br/><a NAME="E14:77_2_1"/>
<font color="Maroon" title="c16">q1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77_2_1"><i><font color="Green" title="E65">A5</font></i></a>, <a class="txt" href="graph_5.html#E3:77_2_1"><i><font color="Green" title="E66">A6</font></i></a>, <a class="ref" href="finseq_3.html#T98" title="FINSEQ_3:th.98">FINSEQ_3:98</a></i>;<br/><a NAME="E15:77_2_1"/><b>then </b>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c16">q1</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c15">p2</font></span>)</span>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77_2"><i><font color="Green" title="E64">A4</font></i></a>, <a class="txt" href="graph_5.html#E5:77_2_1"><i><font color="Green" title="E67">A7</font></i></a>, <a class="txt" href="graph_5.html#E6:77_2_1"><i><font color="Green" title="E68">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E16:77_2_1"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E12:77_2_1"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="graph_5.html#E13:77_2_1"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:77_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:77"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="graph_5.html#E2:77"><i><font color="Green" title="E62">A2</font></i></a>, <a class="txt" href="graph_5.html#E3:77"><i><font color="Green" title="E63">A3</font></i></a>);<br/></i>
<a NAME="E5:77"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">qe</font>]
 
;<br/>
<b>hence </b><a NAME="E6:77"/>
 <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c3">qe</font>,<font color="Maroon" title="c1">W</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="graph_5.html#K10" title="GRAPH_5:func.10">cost</a> <font color="Maroon" title="c4">pe</font>,<font color="Maroon" title="c1">W</font>
 <b>by </b><i><a class="txt" href="graph_5.html#E1:77"><i><font color="Green" title="E61">A1</font></i></a></i>;<br/>


</div>
