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

<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_003.html#V8">nonnegative-weighted</a> <a href="glib_003.html#NM1">WGraph</a>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <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="E1:25"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">W</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">W</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; ( for <font color="Olive">W2</font> being  <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font>  st <font color="Olive">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> holds <br/><font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">W2</font> <a href="glib_003.html#K10">.cost()</a>  ) )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">WC</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font>;<br/>
<a NAME="E3:25"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_001.html#R1">is_Walk_from</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( ( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <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> <font color="Maroon">W</font> ) or <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/><i><font color="Green">E49</font></i>: 
( not <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> &amp; not <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> &amp; <font color="Maroon">m</font> <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> <font color="Maroon">W</font> )
 ;<br/><div class="add"><b>set </b><font color="Maroon">W1</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font>;<br/><b>set </b><font color="Maroon">W3</font> = <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>;<br/><a NAME="E2:25_1_1_1"/><i><font color="Green">E50</font></i>: 
( not 1 is <a href="abian.html#V1">even</a> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="heyting3.html#T1">HEYTING3:1</a>, <a class="ref" href="jordan12.html#T3">JORDAN12:3</a></i>;<br/><a NAME="E3:25_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>;<br/><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> = 
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40">GLIB_001:40</a></i>
;<br/><span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  = 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
;<br/><a NAME="E6:25_1_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T31">GLIB_003:31</a></i>;<br/><a NAME="E7:25_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  = 
<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
<b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> 
<b>by </b><i>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
;<br/><a NAME="E9:25_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T31">GLIB_003:31</a></i>;<br/><a NAME="E10:25_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:25_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E12:25_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_1_1_4"/>
not <font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
 ;<br/><b>then consider </b><font color="Maroon">W2</font> being   <a href="glib_001.html#NM8">DPath</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:25_1_1_1_4"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E4:25_1_1_1_4"/><i><font color="Green">E60</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E5:25_1_1_1_4"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">W2</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">W2</font> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><b>set </b><font color="Maroon">WA</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font>;<br/><b>set </b><font color="Maroon">WB</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span>;<br/><a NAME="E6:25_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> 1,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><a NAME="E7:25_1_1_1_4"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T32">GLIB_001:32</a></i>;<br/><a NAME="E8:25_1_1_1_4"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">W2</font> <a href="glib_001.html#K3">.first()</a> 
 <b>by </b><i>, , , , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><a NAME="E9:25_1_1_1_4"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="glib_003.html#T31">GLIB_003:31</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_001.html#K3">.first()</a>  = 
<font color="Maroon">W2</font> <a href="glib_001.html#K4">.last()</a> 
<b>by </b><i>, , <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
<b>by </b><i>, <a class="ref" href="glib_001.html#T31">GLIB_001:31</a></i>
;<br/><b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="glib_003.html#T31">GLIB_003:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W2</font> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> </span>)</span></span>)</span>

;<br/><a NAME="E12:25_1_1_1_4"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>consider </b><font color="Maroon">WB2</font> being   <a href="glib_001.html#NM14">DPath</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span>;<br/><a NAME="E14:25_1_1_1_4"/>
<font color="Maroon">WB2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> 1,<font color="Maroon">m</font></span>)</span> <a href="glib_001.html#K7">.append</a> <font color="Maroon">W2</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font></span>)</span></span>)</span></span>)</span> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i><a class="ref" href="glib_003.html#T37">GLIB_003:37</a></i>;<br/><a NAME="E15:25_1_1_1_4"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">WB2</font> <a href="glib_003.html#K10">.cost()</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">W</font> <a href="glib_003.html#K10">.cost()</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E16:25_1_1_1_4"/>
<font color="Maroon">WB2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T1" target="_self">Th1</a>, <a class="ref" href="glib_001.html#T161">GLIB_001:161</a></i>;<br/><b>hence </b><a NAME="E17:25_1_1_1_4"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="glib_004.html#E1:25"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E14:25_1_1_1"/>
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">m</font> is <a href="abian.html#V1">even</a> or <font color="Maroon">n</font> is <a href="abian.html#V1">even</a> or  not <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> or  not <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">W</font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_2"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D11">GLIB_001:def 11</a></i>;<br/><a NAME="E3:25_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D23">GLIB_001:def 23</a></i>;<br/><b>hence </b><a NAME="E4:25_1_1_2"/>
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D11">GLIB_001:def 11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:25"/>
<font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font> <a href="glib_004.html#R2" target="_self">is_mincost_DPath_from</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K3">.first()</a> ,<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">m</font>,<font color="Maroon">n</font></span>)</span> <a href="glib_001.html#K4">.last()</a> 
 ;<br/>


</div>
