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<span class="kw">set </span><font color="Maroon" title="c5">FAP</font> =  <a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">CS</font> =  <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">IT1</font>, <font color="Maroon" title="c8">IT2</font> be   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> &amp; <font color="Maroon" title="c7">IT1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c7">IT1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">IT1</font> holds <br/><font color="Maroon" title="c7">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c8">IT2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c8">IT2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">IT2</font> holds <br/><font color="Maroon" title="c8">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) implies <font color="Maroon" title="c7">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">IT2</font> ) &amp; ( not <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> &amp; <font color="Maroon" title="c7">IT1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c8">IT2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">source</font> implies <font color="Maroon" title="c7">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">IT2</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">P1</font>, <font color="Olive" title="b3">P2</font> being  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> $1</span>)</span> &amp; <font color="Olive" title="b2">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b2">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Olive" title="b3">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Olive" title="b1">v</font> &amp; <font color="Olive" title="b3">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">P1</font> holds <br/><font color="Olive" title="b2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b4">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">P2</font> holds <br/><font color="Olive" title="b3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) holds <br/><font color="Olive" title="b2">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">P2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">G0</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_2_1"/><span class="lab"><font color="Green" title="E23">A87</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">set </span><font color="Maroon" title="c11">Gn</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c12">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c13">Next</font> =  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>;<br/><a NAME="E2:30_1_2_1"/><span class="lab"><font color="Green" title="E24">A88</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="glib_005.html#K3" title="GLIB_005:func.3">AP:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#D12" target="_self" title="GLIB_005:def.12">Def12</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_2_1_1_1"/>
(  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or (  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> ) or (  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_1"/>
 <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:30_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1"><span class="lab"><font color="Green" title="E24">A88</font></span></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:30_1_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E23">A87</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A89</font></span>: 
(  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; not <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:30_1_2_1_1_1_2"/>
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A90</font></span>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th4</a></span>;<br/><a NAME="E4:30_1_2_1_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#T5" target="_self" title="GLIB_005:th.5">Th5</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A91</font></span>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E26">A90</font></span></a></span>;<br/><a NAME="E6:30_1_2_1_1_1_2"/>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font> or  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E25">A89</font></span></a>, <a class="ref" href="glib_005.html#D9" target="_self" title="GLIB_005:def.9">Def9</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A92</font></span>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E25">A89</font></span></a>, <a class="ref" href="glib_005.html#D6" target="_self" title="GLIB_005:def.6">Def6</a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def7</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A93</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1"><span class="lab"><font color="Green" title="E24">A88</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E25">A89</font></span></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def10</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A94</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_005.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">P1</font> holds <br/><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">P2</font> holds <br/><font color="Olive" title="b2">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) holds <br/><font color="Olive" title="b1">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">P2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c16">P1</font>, <font color="Maroon" title="c17">P2</font> be   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c16">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c17">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font> holds <br/><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font> holds <br/><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) implies <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A95</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E32">A96</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font>
 <span class="kw">and </span><br/><a NAME="E3:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E33">A97</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E34">A98</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E35">A99</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E36">A100</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font> holds <br/><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E37">A101</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font> holds <br/><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font></span><br/><a NAME="E8:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E38">A102</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E39">A103</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E40">A104</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E41">A105</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_2_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> or <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E30">A94</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A95</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_1_2_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E23">A87</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E32">A96</font></span></a>, <a class="txt" href="#E3:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A97</font></span></a>, <a class="txt" href="#E4:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E34">A98</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E35">A99</font></span></a>, <a class="txt" href="#E6:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A101</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A106</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font></span><br/><div class="add"><a NAME="E2:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A107</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E29">A93</font></span></a>, <a class="txt" href="glib_005.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>;<br/><a NAME="E4:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A108</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E31">A95</font></span></a>, <a class="ref" href="glib_005.html#T7" target="_self" title="GLIB_005:th.7">Th7</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A109</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E25">A89</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E27">A91</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E42">A106</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E41">A105</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A102</font></span></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c16">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A110</font></span>: 
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A103</font></span></a>, <a class="txt" href="#E10:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A104</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E45">A109</font></span></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>;<br/><a NAME="E12:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c17">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></span>;<br/><a NAME="E13:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A111</font></span>: 
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/><a NAME="E14:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A112</font></span>: 
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E15:30_1_2_1_1_1_2_1_1_1_2"/>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A110</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">lenP11</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1, <font color="Maroon" title="c19">lenP21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E48">A112</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E17:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A113</font></span>: 
<font color="Maroon" title="c18">lenP11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E18:30_1_2_1_1_1_2_1_1_1_2"/>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">lenP11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E46">A110</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E19:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A114</font></span>: 
<font color="Maroon" title="c18">lenP11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A113</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E20:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c18">lenP11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 ;<br/><a NAME="E21:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A115</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">lenP11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A102</font></span></a>, <a class="txt" href="#E19:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A114</font></span></a>, <a class="txt" href="#E4:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E44">A108</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">lenP12</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A116</font></span>: 
<font color="Maroon" title="c20">lenP12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">lenP11</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E24:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c18">lenP11</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">and </span><br/><a NAME="E25:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c18">lenP11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">and </span><br/><a NAME="E26:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A117</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">lenP12</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A102</font></span></a>, <a class="txt" href="#E19:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A114</font></span></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></span>;<br/><a NAME="E27:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A118</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">lenP12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E25">A89</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E43">A107</font></span></a>, <a class="txt" href="#E26:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E53">A117</font></span></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">P1A</font> = <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font>;<br/><a NAME="E28:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A119</font></span>: 
<font color="Maroon" title="c20">lenP12</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E49">A113</font></span></a>, <a class="txt" href="#E23:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A116</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E56">A120</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A121</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A122</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A123</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A121</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A124</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:30_1_2_1_1_1_2_1_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A121</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A124</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A125</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_2_1_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">lenP12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="txt" href="#E3:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A123</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_2_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A126</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A122</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:30_1_2_1_1_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A121</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><span class="kw">hence </span><a NAME="E13:30_1_2_1_1_1_2_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E36">A100</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E61">A125</font></span></a>, <a class="txt" href="#E11:30_1_2_1_1_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A126</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E30:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A127</font></span>: 
<font color="Maroon" title="c20">lenP12</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A116</font></span></a></span>;<br/><a NAME="E31:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c20">lenP12</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">lenP11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E52">A116</font></span></a></span>;<br/><a NAME="E32:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A128</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c20">lenP12</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E38">A102</font></span></a>, <a class="txt" href="#E21:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E51">A115</font></span></a>, <a class="txt" href="#E27:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E54">A118</font></span></a>, <a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="txt" href="#E30:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E57">A127</font></span></a>, <a class="ref" href="glib_001.html#T41" title="GLIB_001:th.41">GLIB_001:41</a></span>;<br/><a NAME="E33:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A129</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E33">A97</font></span></a>, <a class="ref" href="glib_005.html#T2" target="_self" title="GLIB_005:th.2">Th2</a></span>;<br/><a NAME="E34:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A130</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">lenP12</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A131</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c20">lenP12</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></span>
<br/>.= 
<font color="Maroon" title="c16">P1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></span>
;<br/><a NAME="E36:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A132</font></span>: 
<font color="Maroon" title="c19">lenP21</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E37:30_1_2_1_1_1_2_1_1_1_2"/>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">lenP21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E47">A111</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E38:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A133</font></span>: 
<font color="Maroon" title="c19">lenP21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E62">A132</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E39:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c19">lenP21</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 ;<br/><a NAME="E40:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A134</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">lenP21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A101</font></span></a>, <a class="txt" href="#E10:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A104</font></span></a>, <a class="txt" href="#E38:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E63">A133</font></span></a>, <a class="txt" href="#E4:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E44">A108</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">lenP22</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E42:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A135</font></span>: 
<font color="Maroon" title="c22">lenP22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">lenP21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E43:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c19">lenP21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">and </span><br/><a NAME="E44:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c19">lenP21</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">and </span><br/><a NAME="E45:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E66">A136</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">lenP22</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A104</font></span></a>, <a class="txt" href="#E38:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E63">A133</font></span></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></span>;<br/><a NAME="E46:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A137</font></span>: 
<font color="Maroon" title="c22">lenP22</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E65">A135</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">P2A</font> = <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font>;<br/><a NAME="E47:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A138</font></span>: 
<font color="Maroon" title="c22">lenP22</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E62">A132</font></span></a>, <a class="txt" href="#E42:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E65">A135</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E69">A139</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_2_1_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="lab"><font color="Green" title="E70">A140</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A141</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="lab"><font color="Green" title="E72">A142</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A140</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A143</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:30_1_2_1_1_1_2_1_1_1_2_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A140</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E73">A143</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A144</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_2_1_1_1_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">lenP22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="txt" href="#E3:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E72">A142</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_2_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A145</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E71">A141</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:30_1_2_1_1_1_2_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A140</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><span class="kw">hence </span><a NAME="E13:30_1_2_1_1_1_2_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E37">A101</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E74">A144</font></span></a>, <a class="txt" href="#E11:30_1_2_1_1_1_2_1_1_1_2_3"><span class="lab"><font color="Green" title="E75">A145</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E49:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A146</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">lenP22</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A147</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c22">lenP22</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></span>
<br/>.= 
<font color="Maroon" title="c17">P2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></span>
;<br/><a NAME="E51:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E72">A148</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">lenP22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E25">A89</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E43">A107</font></span></a>, <a class="txt" href="#E45:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E66">A136</font></span></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></span>;<br/><a NAME="E52:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A149</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E39">A103</font></span></a>, <a class="txt" href="#E49:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E70">A146</font></span></a>, <a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><a NAME="E53:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c22">lenP22</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">lenP21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E65">A135</font></span></a></span>;<br/><a NAME="E54:30_1_2_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A150</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c22">lenP22</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E40">A104</font></span></a>, <a class="txt" href="#E40:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E64">A134</font></span></a>, <a class="txt" href="#E51:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E72">A148</font></span></a>, <a class="txt" href="#E47:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E68">A138</font></span></a>, <a class="txt" href="#E46:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E67">A137</font></span></a>, <a class="ref" href="glib_001.html#T41" title="GLIB_001:th.41">GLIB_001:41</a></span>;<br/><a NAME="E55:30_1_2_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A151</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E35">A99</font></span></a>, <a class="ref" href="glib_005.html#T2" target="_self" title="GLIB_005:th.2">Th2</a></span>;<br/><a NAME="E56:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2_1_1_1_2_1"><span class="lab"><font color="Green" title="E41">A105</font></span></a>, <a class="txt" href="#E27:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E54">A118</font></span></a>, <a class="txt" href="#E34:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E60">A130</font></span></a>, <a class="txt" href="#E28:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E55">A119</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><span class="kw">hence </span><a NAME="E57:30_1_2_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E23">A87</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E28">A92</font></span></a>, <a class="txt" href="#E52:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E73">A149</font></span></a>, <a class="txt" href="#E33:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E59">A129</font></span></a>, <a class="txt" href="#E55:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E75">A151</font></span></a>, <a class="txt" href="#E29:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E56">A120</font></span></a>, <a class="txt" href="#E48:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E69">A139</font></span></a>, <a class="txt" href="#E32:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E58">A128</font></span></a>, <a class="txt" href="#E54:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E74">A150</font></span></a>, <a class="txt" href="#E35:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E61">A131</font></span></a>, <a class="txt" href="#E50:30_1_2_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E71">A147</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:30_1_2_1_1_1_2_1"/>
<font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:30_1_2_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E25">A152</font></span>: 
(  <a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><div class="add"><a NAME="E2:30_1_2_1_1_1_3"/>
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A153</font></span>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th4</a></span>;<br/><a NAME="E4:30_1_2_1_1_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#T5" target="_self" title="GLIB_005:th.5">Th5</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A154</font></span>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E26">A153</font></span></a></span>;<br/><a NAME="E6:30_1_2_1_1_1_3"/>
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font> or  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E25">A152</font></span></a>, <a class="ref" href="glib_005.html#D9" target="_self" title="GLIB_005:def.9">Def9</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A155</font></span>: 
not <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E25">A152</font></span></a>, <a class="ref" href="glib_005.html#D6" target="_self" title="GLIB_005:def.6">Def6</a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def7</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_3"/><span class="lab"><font color="Green" title="E29">A156</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1"><span class="lab"><font color="Green" title="E24">A88</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E25">A152</font></span></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def10</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A157</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_005.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">P1</font> holds <br/><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">P2</font> holds <br/><font color="Olive" title="b2">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) holds <br/><font color="Olive" title="b1">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">P2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c16">P1</font>, <font color="Maroon" title="c17">P2</font> be   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c16">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font> &amp; <font color="Maroon" title="c17">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font> holds <br/><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font> holds <br/><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) implies <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E31">A158</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E32">A159</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font>
 <span class="kw">and </span><br/><a NAME="E3:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E33">A160</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E34">A161</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c15">v</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E35">A162</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E36">A163</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font> holds <br/><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E37">A164</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font> holds <br/><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font></span><br/><a NAME="E8:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E38">A165</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E32">A159</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E39">A166</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E34">A161</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E40">A167</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E34">A161</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E41">A168</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E32">A159</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_3_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_2_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span> or <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E30">A157</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E31">A158</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:30_1_2_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E23">A87</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E32">A159</font></span></a>, <a class="txt" href="#E3:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E33">A160</font></span></a>, <a class="txt" href="#E4:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E34">A161</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E35">A162</font></span></a>, <a class="txt" href="#E6:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E36">A163</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E37">A164</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1_1_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A169</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p4">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font></span><br/><div class="add"><a NAME="E2:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A170</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E29">A156</font></span></a>, <a class="txt" href="glib_005.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>;<br/><a NAME="E4:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A171</font></span>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E31">A158</font></span></a>, <a class="ref" href="glib_005.html#T7" target="_self" title="GLIB_005:th.7">Th7</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A172</font></span>: 
<font color="Maroon" title="c15">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E28">A155</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E27">A154</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E42">A169</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E41">A168</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E38">A165</font></span></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c16">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A173</font></span>: 
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E39">A166</font></span></a>, <a class="txt" href="#E10:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E40">A167</font></span></a>, <a class="txt" href="#E5:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E45">A172</font></span></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></span>;<br/><a NAME="E12:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c17">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></span>;<br/><a NAME="E13:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A174</font></span>: 
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></span>;<br/><a NAME="E14:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A175</font></span>: 
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E15:30_1_2_1_1_1_3_1_1_1_2"/>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E46">A173</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">lenP11</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1, <font color="Maroon" title="c19">lenP21</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E48">A175</font></span></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></span>;<br/><a NAME="E17:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A176</font></span>: 
<font color="Maroon" title="c18">lenP11</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E18:30_1_2_1_1_1_3_1_1_1_2"/>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">lenP11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E46">A173</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E19:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A177</font></span>: 
<font color="Maroon" title="c18">lenP11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E49">A176</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E20:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c18">lenP11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 ;<br/><a NAME="E21:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A178</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">lenP11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E36">A163</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E38">A165</font></span></a>, <a class="txt" href="#E19:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E50">A177</font></span></a>, <a class="txt" href="#E4:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E44">A171</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">lenP12</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A179</font></span>: 
<font color="Maroon" title="c20">lenP12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">lenP11</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E24:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c18">lenP11</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">and </span><br/><a NAME="E25:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c18">lenP11</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">and </span><br/><a NAME="E26:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A180</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">lenP12</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E38">A165</font></span></a>, <a class="txt" href="#E19:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E50">A177</font></span></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></span>;<br/><a NAME="E27:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A181</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">lenP12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E25">A152</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E28">A155</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E43">A170</font></span></a>, <a class="txt" href="#E26:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E53">A180</font></span></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">P1A</font> = <font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font>;<br/><a NAME="E28:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A182</font></span>: 
<font color="Maroon" title="c20">lenP12</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E49">A176</font></span></a>, <a class="txt" href="#E23:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E52">A179</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E56">A183</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_3_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A184</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A185</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A186</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A184</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A187</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:30_1_2_1_1_1_3_1_1_1_2_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A184</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A187</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A188</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_3_1_1_1_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">lenP12</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="txt" href="#E3:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A186</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_3_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A189</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A185</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:30_1_2_1_1_1_3_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A184</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><span class="kw">hence </span><a NAME="E13:30_1_2_1_1_1_3_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E36">A163</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E61">A188</font></span></a>, <a class="txt" href="#E11:30_1_2_1_1_1_3_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A189</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E30:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A190</font></span>: 
<font color="Maroon" title="c20">lenP12</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E52">A179</font></span></a></span>;<br/><a NAME="E31:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c20">lenP12</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">lenP11</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E52">A179</font></span></a></span>;<br/><a NAME="E32:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A191</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c20">lenP12</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E38">A165</font></span></a>, <a class="txt" href="#E21:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E51">A178</font></span></a>, <a class="txt" href="#E27:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E54">A181</font></span></a>, <a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="txt" href="#E30:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E57">A190</font></span></a>, <a class="ref" href="glib_001.html#T41" title="GLIB_001:th.41">GLIB_001:41</a></span>;<br/><a NAME="E33:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A192</font></span>: 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E33">A160</font></span></a>, <a class="ref" href="glib_005.html#T2" target="_self" title="GLIB_005:th.2">Th2</a></span>;<br/><a NAME="E34:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A193</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">lenP12</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A194</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c20">lenP12</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">P1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></span>
<br/>.= 
<font color="Maroon" title="c16">P1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></span>
;<br/><a NAME="E36:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A195</font></span>: 
<font color="Maroon" title="c19">lenP21</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E37:30_1_2_1_1_1_3_1_1_1_2"/>
3 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">lenP21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E47">A174</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><a NAME="E38:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A196</font></span>: 
<font color="Maroon" title="c19">lenP21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E62">A195</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E39:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c19">lenP21</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 ;<br/><a NAME="E40:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A197</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">lenP21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E37">A164</font></span></a>, <a class="txt" href="#E10:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E40">A167</font></span></a>, <a class="txt" href="#E38:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E63">A196</font></span></a>, <a class="txt" href="#E4:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E44">A171</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">lenP22</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E42:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A198</font></span>: 
<font color="Maroon" title="c22">lenP22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">lenP21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E43:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c19">lenP21</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">and </span><br/><a NAME="E44:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c19">lenP21</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">and </span><br/><a NAME="E45:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E66">A199</font></span>: 
 <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">lenP22</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E40">A167</font></span></a>, <a class="txt" href="#E38:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E63">A196</font></span></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></span>;<br/><a NAME="E46:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A200</font></span>: 
<font color="Maroon" title="c22">lenP22</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E65">A198</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">P2A</font> = <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font>;<br/><a NAME="E47:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A201</font></span>: 
<font color="Maroon" title="c22">lenP22</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E62">A195</font></span></a>, <a class="txt" href="#E42:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E65">A198</font></span></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E69">A202</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1_3_1_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">n</font> be   <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="lab"><font color="Green" title="E70">A203</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span><br/><a NAME="E2:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A204</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="lab"><font color="Green" title="E72">A205</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A203</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A206</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:30_1_2_1_1_1_3_1_1_1_2_3"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A203</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E73">A206</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E8:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A207</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E9:30_1_2_1_1_1_3_1_1_1_2_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">lenP22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></span>;<br/><a NAME="E10:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="txt" href="#E3:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E72">A205</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:30_1_2_1_1_1_3_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A208</font></span>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E71">A204</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:30_1_2_1_1_1_3_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A203</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><span class="kw">hence </span><a NAME="E13:30_1_2_1_1_1_3_1_1_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E37">A164</font></span></a>, <a class="txt" href="#E8:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E74">A207</font></span></a>, <a class="txt" href="#E11:30_1_2_1_1_1_3_1_1_1_2_3"><span class="lab"><font color="Green" title="E75">A208</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E49:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E70">A209</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">lenP22</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A210</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c22">lenP22</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></span>
<br/>.= 
<font color="Maroon" title="c17">P2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></span>
;<br/><a NAME="E51:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E72">A211</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">lenP22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E25">A152</font></span></a>, <a class="txt" href="#E7:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E28">A155</font></span></a>, <a class="txt" href="#E2:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E43">A170</font></span></a>, <a class="txt" href="#E45:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E66">A199</font></span></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></span>;<br/><a NAME="E52:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A212</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E39">A166</font></span></a>, <a class="txt" href="#E49:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E70">A209</font></span></a>, <a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><a NAME="E53:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c22">lenP22</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">lenP21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E65">A198</font></span></a></span>;<br/><a NAME="E54:30_1_2_1_1_1_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A213</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c22">lenP22</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">P2</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c15">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E40">A167</font></span></a>, <a class="txt" href="#E40:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E64">A197</font></span></a>, <a class="txt" href="#E51:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E72">A211</font></span></a>, <a class="txt" href="#E47:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E68">A201</font></span></a>, <a class="txt" href="#E46:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E67">A200</font></span></a>, <a class="ref" href="glib_001.html#T41" title="GLIB_001:th.41">GLIB_001:41</a></span>;<br/><a NAME="E55:30_1_2_1_1_1_3_1_1_1_2"/><span class="lab"><font color="Green" title="E75">A214</font></span>: 
<font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c22">lenP22</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E35">A162</font></span></a>, <a class="ref" href="glib_005.html#T2" target="_self" title="GLIB_005:th.2">Th2</a></span>;<br/><a NAME="E56:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c16">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c20">lenP12</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p2">(<span class="default"><a href="glib_005.html#K2" title="GLIB_005:func.2">AP:NextBestEdges</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E41">A168</font></span></a>, <a class="txt" href="#E27:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E54">A181</font></span></a>, <a class="txt" href="#E34:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E60">A193</font></span></a>, <a class="txt" href="#E28:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E55">A182</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><span class="kw">hence </span><a NAME="E57:30_1_2_1_1_1_3_1_1_1_2"/>
<font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_1"><span class="lab"><font color="Green" title="E23">A87</font></span></a>, <a class="txt" href="#E1:30_1_2_1_1_1_3"><span class="lab"><font color="Green" title="E25">A152</font></span></a>, <a class="txt" href="#E52:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E73">A212</font></span></a>, <a class="txt" href="#E33:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E59">A192</font></span></a>, <a class="txt" href="#E55:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E75">A214</font></span></a>, <a class="txt" href="#E29:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E56">A183</font></span></a>, <a class="txt" href="#E48:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E69">A202</font></span></a>, <a class="txt" href="#E32:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E58">A191</font></span></a>, <a class="txt" href="#E54:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E74">A213</font></span></a>, <a class="txt" href="#E35:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E61">A194</font></span></a>, <a class="txt" href="#E50:30_1_2_1_1_1_3_1_1_1_2"><span class="lab"><font color="Green" title="E71">A210</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E13:30_1_2_1_1_1_3_1"/>
<font color="Maroon" title="c16">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c17">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:30_1_2_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:30_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:30_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A215</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c10">v</font> &amp; <font color="Olive" title="b1">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c10">v</font> &amp; <font color="Olive" title="b2">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">P1</font> holds <br/><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">P2</font> holds <br/><font color="Olive" title="b2">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) holds <br/><font color="Olive" title="b1">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b2">P2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">P1</font>, <font color="Maroon" title="c12">P2</font> be   <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a>  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp; <font color="Maroon" title="c11">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c10">v</font> &amp; <font color="Maroon" title="c11">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; <font color="Maroon" title="c12">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c10">v</font> &amp; <font color="Maroon" title="c12">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">P1</font> holds <br/><font color="Maroon" title="c11">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">P2</font> holds <br/><font color="Maroon" title="c12">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) implies <font color="Maroon" title="c11">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">P2</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30_1_2_2"/><span class="lab"><font color="Green" title="E24">A216</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="glib_005.html#K4" title="GLIB_005:func.4">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:30_1_2_2"/><span class="lab"><font color="Green" title="E25">A217</font></span>: 
<font color="Maroon" title="c11">P1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">and </span><br/><a NAME="E3:30_1_2_2"/>
<font color="Maroon" title="c11">P1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_2_2"/><span class="lab"><font color="Green" title="E26">A218</font></span>: 
<font color="Maroon" title="c12">P2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c10">v</font>
 <span class="kw">and </span><br/><a NAME="E5:30_1_2_2"/>
<font color="Maroon" title="c12">P2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 <span class="kw">and </span><br/><a NAME="E6:30_1_2_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">P1</font> holds <br/><font color="Maroon" title="c11">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:30_1_2_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">P2</font> holds <br/><font color="Maroon" title="c12">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">P2</font></span><br/><a NAME="E8:30_1_2_2"/>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">source</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_2_2"><span class="lab"><font color="Green" title="E24">A216</font></span></a>, <a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th4</a></span>;<br/><a NAME="E9:30_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A219</font></span>: 
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:30_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A220</font></span>: 
<font color="Maroon" title="c11">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_2"><span class="lab"><font color="Green" title="E25">A217</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E11:30_1_2_2"/><span class="lab"><font color="Green" title="E29">A221</font></span>: 
<font color="Maroon" title="c12">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2_2"><span class="lab"><font color="Green" title="E26">A218</font></span></a>, <a class="txt" href="#E9:30_1_2_2"><span class="lab"><font color="Green" title="E27">A219</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E12:30_1_2_2"/><span class="lab"><font color="Green" title="E30">A222</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E13:30_1_2_2"/>
<font color="Maroon" title="c11">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_2_2"><span class="lab"><font color="Green" title="E25">A217</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E14:30_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2_2"><span class="lab"><font color="Green" title="E28">A220</font></span></a>, <a class="txt" href="#E12:30_1_2_2"><span class="lab"><font color="Green" title="E30">A222</font></span></a>, <a class="ref" href="glib_001.html#D29" title="GLIB_001:def.29">GLIB_001:def 29</a></span>;<br/><a NAME="E15:30_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A223</font></span>: 
<font color="Maroon" title="c11">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:30_1_2_2"><span class="lab"><font color="Green" title="E28">A220</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><a NAME="E16:30_1_2_2"/><span class="lab"><font color="Green" title="E32">A224</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E17:30_1_2_2"/>
<font color="Maroon" title="c12">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">P2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_2_2"><span class="lab"><font color="Green" title="E26">A218</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E18:30_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2_2"><span class="lab"><font color="Green" title="E29">A221</font></span></a>, <a class="txt" href="#E16:30_1_2_2"><span class="lab"><font color="Green" title="E32">A224</font></span></a>, <a class="ref" href="glib_001.html#D29" title="GLIB_001:def.29">GLIB_001:def 29</a></span>;<br/><span class="kw">hence </span><a NAME="E19:30_1_2_2"/>
<font color="Maroon" title="c11">P1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">P2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_1_2_2"><span class="lab"><font color="Green" title="E29">A221</font></span></a>, <a class="txt" href="#E15:30_1_2_2"><span class="lab"><font color="Green" title="E31">A223</font></span></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:30_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A225</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E5:30_1_2"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E4:30_1_2"><span class="lab"><font color="Green" title="E24">A225</font></span></a>, <a class="txt" href="#E2:30_1_2"><span class="lab"><font color="Green" title="E23">A215</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:30_1_2"/>
( <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> &amp; <font color="Maroon" title="c7">IT1</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c7">IT1</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">IT1</font> holds <br/><font color="Maroon" title="c7">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">IT1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon" title="c8">IT2</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c8">IT2</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">IT2</font> holds <br/><font color="Maroon" title="c8">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">IT2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> ) implies <font color="Maroon" title="c7">IT1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c8">IT2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> &amp; <font color="Maroon" title="c7">IT1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c8">IT2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">source</font> implies <font color="Maroon" title="c7">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">IT2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:30_1_2"/>
( not <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c3">source</font></span>)</span> &amp; <font color="Maroon" title="c7">IT1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c8">IT2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_001.html#K1" title="GLIB_001:func.1">.walkOf</a> <font color="Maroon" title="c3">source</font> implies <font color="Maroon" title="c7">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">IT2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
