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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>;<br/><b>let </b><font color="Maroon" title="c2">W1</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">W2</font> be    <a href="glib_001.html#M4" title="GLIB_001:mode.4">Subwalk</a> of <font color="Maroon" title="c2">W1</font>;<br/>





<b>consider </b><font color="Maroon" title="c4">ws</font> being    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a> <b> such that </b><br/><a NAME="E2:175"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_003.html#K1" title="GLIB_003:func.1">Seq</a> <font color="Maroon" title="c4">ws</font>
 <b>by </b><i><a class="ref" href="glib_003.html#T25" target="_self" title="GLIB_003:th.25">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">ws'</font> = <font color="Maroon" title="c4">ws</font> as    <a href="graph_2.html#M1" title="GRAPH_2:mode.1">FinSubsequence</a> of <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a>  ;<br/>
<a NAME="E4:175"/>
for <font color="Olive" title="b1">i</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">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a> </span>)</span> holds <br/>0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font>
 
<b>by </b><i><a class="ref" href="glib_003.html#T35" target="_self" title="GLIB_003:th.35">Th35</a></i>;<br/>
<a NAME="E5:175"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a> </span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="glib_003.html#E2:175"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="glib_003.html#T2" target="_self" title="GLIB_003:th.2">Th2</a></i>;<br/>
<a NAME="E6:175"/><b>then </b>
<font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K9" title="GLIB_003:func.9">.weightSeq()</a> </span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:175"/>
<font color="Maroon" title="c3">W2</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">W1</font> <a href="glib_003.html#K10" title="GLIB_003:func.10">.cost()</a> 
 ;<br/>


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