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<b>let </b><font color="Maroon">G</font> be  <a href="glib_003.html#V7" target="_self">real-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">W1</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">W2</font> be    <a href="glib_001.html#M4">Subwalk</a> of <font color="Maroon">W1</font>;<br/>





<b>consider </b><font color="Maroon">es</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> <b> such that </b><br/><a NAME="E2:166"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">W2</font> <a href="glib_001.html#K12">.edgeSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D32">GLIB_001:def 32</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">ws</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:166"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ws</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</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="relat_1.html#K1">dom</a> <font color="Maroon">es</font> holds <br/><font color="Maroon">ws</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:166"/><i><font color="Green">E51</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ws</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<a NAME="E7:166"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">es</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> 
 
<b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">ws</font> = <font color="Maroon">ws</font> as   <a href="finseq_1.html#NM4">FinSubsequence</a> <b>by </b><i>, <a class="ref" href="finseq_1.html#D12">FINSEQ_1:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="166_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:166_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ws</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:166_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E4:166_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font> &amp; <font color="Maroon">ws</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E5:166_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">es</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E6:166_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E8:166_1"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:166_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:166_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:166_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E12:166_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:166_1"/>
<span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E14:166_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> 
 <b>by </b><i>, , , , , <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:166"/><b>then </b>
<font color="Maroon">ws</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ws</font> = <font color="Maroon">ws</font> as    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">W1</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a>  <b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">ws</font>
;<br/>

<i><font color="Green">E70</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">es</font></span>)</span> = 
 <a href="card_1.html#K4">card</a> <font color="Maroon">es</font>
<b>by </b><i><a class="ref" href="glib_001.html#T5">GLIB_001:5</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">ws</font></span>)</span>
<b>by </b><i>, <a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <font color="Maroon">ws</font>
<b>by </b><i><a class="ref" href="pre_circ.html#T21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_001.html#T5">GLIB_001:5</a></i>
;<br/>
<a NAME="E13:166"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="166_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:166_3"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span> )
 ;<br/><a NAME="E2:166_3"/><i><font color="Green">E75</font></i>: 
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ws</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E3:166_3"/><i><font color="Green">E76</font></i>: 
 <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">es</font> <a href="hidden.html#R1">=</a> <font color="Maroon">es</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/><a NAME="E4:166_3"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/><a NAME="E5:166_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E6:166_3"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E7:166_3"/><b>then </b><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:166_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">es</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">ws</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">es</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">es</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_003.html#K5" target="_self">the_Weight_of</a> <font color="Maroon">G</font></span>)</span> <a href="seq_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">es</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/><b>hence </b><a NAME="E10:166_3"/>
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:166"/>
<font color="Maroon">W2</font> <a href="glib_003.html#K9" target="_self">.weightSeq()</a>  <a href="hidden.html#R1">=</a>  <a href="glib_003.html#K1" target="_self">Seq</a> <font color="Maroon">ws</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div>
