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<div class="add">

<b>let </b><font color="Maroon">IT1</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>, <font color="Maroon">IT2</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/>

<b>set </b><font color="Maroon">FAP</font> =  <a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>;<br/>
<b>set </b><font color="Maroon">CS</font> =  <a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">P1</font>, <font color="Olive">P2</font> being <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Olive">P1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Olive">v</font> &amp; <font color="Olive">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Olive">v</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P1</font> holds <br/><font color="Olive">P1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Olive">P2</font> holds <br/><font color="Olive">P2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">P2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) holds <br/><font color="Olive">P1</font> <a href="hidden.html#R1">=</a> <font color="Olive">P2</font>;<br/>
<b>set </b><font color="Maroon">G0</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P1</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>, <font color="Maroon">P2</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:48_1_2_1"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">P1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> holds <br/><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> holds <br/><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/><a NAME="E2:48_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_1_2_1"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">source</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:48_1_2_1"/><b>then </b><i><font color="Green">E211</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">P1</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">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">P2</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">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#T18">GLIB_001:18</a></i>;<br/><a NAME="E5:48_1_2_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E6:48_1_2_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D29">GLIB_001:def 29</a></i>;<br/><a NAME="E7:48_1_2_1"/><b>then </b>
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">v</font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E8:48_1_2_1"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:48_1_2"/><b>then </b><i><font color="Green">E212</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2"><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="E1:48_1_2_2"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>set </b><font color="Maroon">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><a NAME="E2:48_1_2_2"/><i><font color="Green">E214</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_005.html#K4" target="_self">AP:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:48_1_2_2"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">G</font> <a href="glib_000.html#R5">==</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">Next</font> =  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>;<br/><b>set </b><font color="Maroon">e</font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_2_1_1"/>
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or (  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) or (  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_1"/>
 <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:48_1_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:48_1_2_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_2"/><i><font color="Green">E216</font></i>: 
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:48_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E217</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_1_2_2_1_1_2"/><i><font color="Green">E218</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:48_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E219</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E220</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_2"/>
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:48_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E221</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E8:48_1_2_2_1_1_2"/>
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:48_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E10:48_1_2_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E11:48_1_2_2_1_1_2"/><b>then </b><i><font color="Green">E223</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P1</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>, <font color="Maroon">P2</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:48_1_2_2_1_1_2_1"/><i><font color="Green">E224</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">P1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> holds <br/><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> holds <br/><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/><a NAME="E2:48_1_2_2_1_1_2_1"/><b>then </b><i><font color="Green">E225</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">P1</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">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">P2</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">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</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="48_1_2_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_2_1_1_2_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_2_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_2_1_1_2_1_1_1_1"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E227</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:48_1_2_2_1_1_2_1_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">source</font>
 <b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P1</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P2</font> <a href="glib_001.html#K4">.last()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">P1</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P1</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">P2</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P2</font> <a href="glib_001.html#K4">.last()</a>  )
 <b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( not <font color="Maroon">P1</font> is <a href="glib_001.html#V3">trivial</a> &amp; not <font color="Maroon">P2</font> is <a href="glib_001.html#V3">trivial</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><a NAME="E7:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E228</font></i>: 
( 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E8:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>then reconsider </b><font color="Maroon">lenP11</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1, <font color="Maroon">lenP21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E10:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E231</font></i>: 
( 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP11</font> &amp; 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP21</font> )
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">lenP11</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP21</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 ;<br/><a NAME="E12:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E233</font></i>: 
( <font color="Maroon">lenP11</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; <font color="Maroon">lenP21</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E13:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E234</font></i>: 
( <font color="Maroon">lenP11</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP21</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:48_1_2_2_1_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_003.html#T46">GLIB_003:46</a></i>;<br/><a NAME="E15:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E16:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E235</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP11</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP21</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , </i>;<br/><b>then consider </b><font color="Maroon">lenP12</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">lenP12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP11</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">lenP11</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP11</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP12</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T10">GLIB_001:10</a></i>;<br/><a NAME="E19:48_1_2_2_1_1_2_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E20:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E21:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E238</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP12</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><b>consider </b><font color="Maroon">lenP22</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E240</font></i>: 
( <font color="Maroon">lenP22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP21</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">lenP21</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">lenP21</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP22</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T10">GLIB_001:10</a></i>;<br/><a NAME="E24:48_1_2_2_1_1_2_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E25:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E26:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E241</font></i>: 
<font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP22</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , , <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><b>set </b><font color="Maroon">P1A</font> = <font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font>;<br/><b>set </b><font color="Maroon">P2A</font> = <font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font>;<br/><a NAME="E27:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E244</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP12</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP22</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E28:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">lenP12</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP22</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E29:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E246</font></i>: 
( <font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, <a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a></i>;<br/><div><i><font color="Green">E247</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_2_2_1_1_2_1_1_1_2_1"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span>
 ;<br/><a NAME="E2:48_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E249</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">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:48_1_2_2_1_1_2_1_1_1_2_1"/><i><font color="Green">E250</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E4:48_1_2_2_1_1_2_1_1_1_2_1"/>
<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">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:48_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b><i><font color="Green">E251</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E8:48_1_2_2_1_1_2_1_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP12</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T46">GLIB_001:46</a></i>;<br/><a NAME="E9:48_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:48_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_2_2_1_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E252</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1_1_2_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_2_2_1_1_2_1_1_1_2_2"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span>
 ;<br/><a NAME="E2:48_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b><i><font color="Green">E254</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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:48_1_2_2_1_1_2_1_1_1_2_2"/><i><font color="Green">E255</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E4:48_1_2_2_1_1_2_1_1_1_2_2"/>
<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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:48_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b><i><font color="Green">E256</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E8:48_1_2_2_1_1_2_1_1_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP22</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T46">GLIB_001:46</a></i>;<br/><a NAME="E9:48_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:48_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_2_2_1_1_2_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E32:48_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green">E257</font></i>: 
( <font color="Maroon">lenP12</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lenP11</font> &amp; <font color="Maroon">lenP22</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lenP21</font> )
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E33:48_1_2_2_1_1_2_1_1_1_2"/>
( <font color="Maroon">lenP12</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP22</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E34:48_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E258</font></i>: 
( <font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP12</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP22</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, , <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="glib_001.html#T41">GLIB_001:41</a></i>;<br/><i><font color="Green">E259</font></i>: <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP12</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span></span>)</span> = 
<font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon">P1</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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP22</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span></span>)</span> = 
<font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40">GLIB_001:40</a></i>
;<br/><b>hence </b><a NAME="E37:48_1_2_2_1_1_2_1_1_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_2_2_1_1_2_1"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:48_1_2_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_3"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_3"/><i><font color="Green">E260</font></i>: 
(  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  )
 ;<br/><div class="add"><a NAME="E2:48_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E261</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K13">.labelVertex</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E3:48_1_2_2_1_1_3"/><i><font color="Green">E262</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:48_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E263</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> </span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="glib_003.html#T59">GLIB_003:59</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_3"/>
(  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" target="_self">is_forward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> or  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" target="_self">is_backward_labeling_in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E7:48_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E265</font></i>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i>, , </i>;<br/><a NAME="E8:48_1_2_2_1_1_3"/>
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">source</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:48_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E266</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E10:48_1_2_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> 0</span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><a NAME="E11:48_1_2_2_1_1_3"/><b>then </b><i><font color="Green">E267</font></i>: 
<font color="Maroon">source</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">P1</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>, <font color="Maroon">P2</font> be  <a href="glib_001.html#V6">vertex-distinct</a> <a href="glib_005.html#V7" target="_self">augmenting</a> <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:48_1_2_2_1_1_3_1"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">P1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">v</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> holds <br/><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> holds <br/><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/><a NAME="E2:48_1_2_2_1_1_3_1"/><b>then </b><i><font color="Green">E269</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">P1</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">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">source</font> &amp; <font color="Maroon">P2</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">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</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="48_1_2_2_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_2_1_1_3_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_3_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_2_1_1_3_1_1_1_1"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_2_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E271</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11">the_Target_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:48_1_2_2_1_1_3_1_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">source</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, , </i>;<br/><a NAME="E4:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P1</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P2</font> <a href="glib_001.html#K4">.last()</a>  )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D7">GLIB_001:def 7</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( <font color="Maroon">P1</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P1</font> <a href="glib_001.html#K4">.last()</a>  &amp; <font color="Maroon">P2</font> <a href="glib_001.html#K3">.first()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P2</font> <a href="glib_001.html#K4">.last()</a>  )
 <b>by </b><i><a class="ref" href="glib_001.html#D6">GLIB_001:def 6</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( not <font color="Maroon">P1</font> is <a href="glib_001.html#V3">trivial</a> &amp; not <font color="Maroon">P2</font> is <a href="glib_001.html#V3">trivial</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#T128">GLIB_001:128</a></i>;<br/><a NAME="E7:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E272</font></i>: 
( 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T126">GLIB_001:126</a></i>;<br/><a NAME="E8:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>then reconsider </b><font color="Maroon">lenP11</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1, <font color="Maroon">lenP21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 as  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E10:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E273</font></i>: 
( 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP11</font> &amp; 3 <a href="real_1.html#K5">-</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP21</font> )
 <b>by </b><i>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E11:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">lenP11</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP21</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 ;<br/><a NAME="E12:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E275</font></i>: 
( <font color="Maroon">lenP11</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 &amp; <font color="Maroon">lenP21</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 )
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E13:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E276</font></i>: 
( <font color="Maroon">lenP11</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP21</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:48_1_2_2_1_1_3_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="glib_003.html#T46">GLIB_003:46</a></i>;<br/><a NAME="E15:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E16:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E277</font></i>: 
( <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP11</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP21</font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i>, , , </i>;<br/><b>then consider </b><font color="Maroon">lenP12</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E18:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">lenP12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP11</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">lenP11</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP11</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP12</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T10">GLIB_001:10</a></i>;<br/><a NAME="E19:48_1_2_2_1_1_3_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E20:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E21:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E279</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP12</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><b>consider </b><font color="Maroon">lenP22</font> being  <a href="abian.html#V1">odd</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">lenP22</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP21</font> <a href="real_1.html#K5">-</a> 1 &amp; <font color="Maroon">lenP21</font> <a href="real_1.html#K5">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> &amp; <font color="Maroon">lenP21</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font> &amp;  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP22</font>,<font color="Maroon">v</font>,<font color="Maroon">G</font> )
 <b>by </b><i>, , , <a class="ref" href="glib_001.html#T10">GLIB_001:10</a></i>;<br/><a NAME="E24:48_1_2_2_1_1_3_1_1_1_2"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<span class="p1">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="glib_000.html#D15">GLIB_000:def 15</a></i>;<br/><a NAME="E25:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1">Joins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font>,<font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="glib_000.html#T91">GLIB_000:91</a></i>;<br/><a NAME="E26:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E281</font></i>: 
<font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lenP22</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, , , <a class="ref" href="glib_000.html#T18">GLIB_000:18</a></i>;<br/><b>set </b><font color="Maroon">P1A</font> = <font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font>;<br/><b>set </b><font color="Maroon">P2A</font> = <font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font>;<br/><a NAME="E27:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E282</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP12</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lenP22</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1">HEYTING3:1</a></i>;<br/><a NAME="E28:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E283</font></i>: 
( <font color="Maroon">lenP12</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP22</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a>, <a class="ref" href="glib_005.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E29:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E284</font></i>: 
( <font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_001.html#T38">GLIB_001:38</a>, , , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a></i>;<br/><div><i><font color="Green">E285</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1_1_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_2_2_1_1_3_1_1_1_2_1"/><i><font color="Green">E286</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span>
 ;<br/><a NAME="E2:48_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E287</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">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:48_1_2_2_1_1_3_1_1_1_2_1"/><i><font color="Green">E288</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E4:48_1_2_2_1_1_3_1_1_1_2_1"/>
<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">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:48_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b><i><font color="Green">E289</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E8:48_1_2_2_1_1_3_1_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP12</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T46">GLIB_001:46</a></i>;<br/><a NAME="E9:48_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:48_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_2_2_1_1_3_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><i><font color="Green">E290</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2_1_1_3_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be  <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:48_1_2_2_1_1_3_1_1_1_2_2"/><i><font color="Green">E291</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span>
 ;<br/><a NAME="E2:48_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b><i><font color="Green">E292</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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:48_1_2_2_1_1_3_1_1_1_2_2"/><i><font color="Green">E293</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E4:48_1_2_2_1_1_3_1_1_1_2_2"/>
<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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E5:48_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:48_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:48_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b><i><font color="Green">E294</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T47">GLIB_001:47</a></i>;<br/><a NAME="E8:48_1_2_2_1_1_3_1_1_1_2_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lenP22</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T46">GLIB_001:46</a></i>;<br/><a NAME="E9:48_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:48_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:48_1_2_2_1_1_3_1_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E32:48_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green">E295</font></i>: 
( <font color="Maroon">lenP12</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lenP11</font> &amp; <font color="Maroon">lenP22</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lenP21</font> )
 <b>by </b><i>, <a class="ref" href="glib_005.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E33:48_1_2_2_1_1_3_1_1_1_2"/>
( <font color="Maroon">lenP12</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">lenP22</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> )
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self">Def2</a>, </i>;<br/><a NAME="E34:48_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green">E296</font></i>: 
( <font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP12</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font> &amp; <font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP22</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10">the_Source_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K3" target="_self">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" target="_self">AP:CompSeq</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_004.html#K5">.</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">v</font> )
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D3" target="_self">Def3</a>, , <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, , <a class="ref" href="glib_001.html#T41">GLIB_001:41</a></i>;<br/><i><font color="Green">E297</font></i>: <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP12</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP12</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span></span>)</span> = 
<font color="Maroon">P1</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon">P1</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">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">lenP22</font></span>)</span> <a href="glib_001.html#K7">.append</a> <span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <font color="Maroon">lenP22</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span></span>)</span> = 
<font color="Maroon">P2</font> <a href="glib_001.html#K8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D4" target="_self">Def4</a>, <a class="ref" href="glib_005.html#D5" target="_self">Def5</a>, <a class="ref" href="glib_001.html#T39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40">GLIB_001:40</a></i>
;<br/><b>hence </b><a NAME="E37:48_1_2_2_1_1_3_1_1_1_2"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , <a class="ref" href="glib_005.html#D6" target="_self">Def6</a>, , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_2_2_1_1_3_1"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:48_1_2_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:48_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:48_1_2"/><b>then </b><i><font color="Green">E298</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E5:48_1_2"/><i><font color="Green">E299</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>thus </b><a NAME="E6:48_1_2"/>
( <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">IT1</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">IT1</font> holds <br/><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">IT1</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">IT2</font> <a href="glib_001.html#R1">is_Walk_from</a> <font color="Maroon">source</font>,<font color="Maroon">sink</font> &amp; ( for <font color="Olive">n</font> being <a href="abian.html#V1">even</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">IT2</font> holds <br/><font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">IT2</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) implies <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E7:48_1_2"/>
( not <font color="Maroon">sink</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" target="_self">AP:FindAugPath</a> <font color="Maroon">G</font>,<font color="Maroon">source</font></span>)</span> <a href="glib_003.html#K14">.labeledV()</a>  &amp; <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">source</font> &amp; <font color="Maroon">IT2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="glib_001.html#K1">.walkOf</a> <font color="Maroon">source</font> implies <font color="Maroon">IT1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT2</font> )
 ;<br/>


</div>
