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<b>let </b><font color="Maroon" title="c5">IT1</font>, <font color="Maroon" title="c6">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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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="c5">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="c5">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">IT1</font> holds <br/><font color="Maroon" title="c5">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="c5">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="c6">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="c6">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">IT2</font> holds <br/><font color="Maroon" title="c6">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="c6">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="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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#K4" title="RELSET_1:func.4">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="c5">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="c6">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="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> ) )</span><br/>

<b>set </b><font color="Maroon" title="c7">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/>
<b>set </b><font color="Maroon" title="c8">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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/>
<b>set </b><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"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/><b>assume </b><a NAME="E1:30_1_2_1"/><i><font color="Green" title="E24">A59</font></i>: 
( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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="E2:30_1_2_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th3</a></i>;<br/><a NAME="E3:30_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:30_1_2_1"/><b>then </b><i><font color="Green" title="E25">A60</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_1"><i><font color="Green" title="E24">A59</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E5:30_1_2_1"/>
( 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> &amp; 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> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E6:30_1_2_1"/><b>then </b>
(  <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 &amp;  <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 )
 <b>by </b><i><a class="txt" href="#E4:30_1_2_1"><i><font color="Green" title="E25">A60</font></i></a>, <a class="ref" href="glib_001.html#D29" title="GLIB_001:def.29">GLIB_001:def 29</a></i>;<br/><a NAME="E7:30_1_2_1"/><b>then </b>
( <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> &amp; <font color="Maroon" title="c12">P2</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> )
 <b>by </b><i><a class="txt" href="#E4:30_1_2_1"><i><font color="Green" title="E25">A60</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E8:30_1_2_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:30_1_2"/><b>then </b><i><font color="Green" title="E24">A61</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <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_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:30_1_2_2"/><i><font color="Green" title="E25">A62</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>set </b><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/><b>set </b><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/><a NAME="E2:30_1_2_2"/><i><font color="Green" title="E26">A63</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="glib_005.html#D12" target="_self" title="GLIB_005:def.12">Def15</a></i>;<br/><b>set </b><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/><b>set </b><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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_2_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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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_2_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2"><i><font color="Green" title="E26">A63</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def14</a></i>;<br/><b>hence </b><a NAME="E3:30_1_2_2_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]
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2"><i><font color="Green" title="E25">A62</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_2_1_1_2"/><i><font color="Green" title="E27">A65</font></i>: 
(  <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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_2"/><b>then </b><i><font color="Green" title="E28">A66</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2"><i><font color="Green" title="E26">A63</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def14</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_2"/><i><font color="Green" title="E29">A68</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="txt" href="glib_005.html#E1"><i><font color="Green" title="E1">Tw0</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2"><i><font color="Green" title="E28">A66</font></i></a></i>;<br/><a NAME="E4:30_1_2_2_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> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2"><i><font color="Green" title="E27">A65</font></i></a>, <a class="ref" href="glib_005.html#D9" target="_self" title="GLIB_005:def.9">Def13</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E30">A69</font></i>: 
<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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2"><i><font color="Green" title="E27">A65</font></i></a>, <a class="ref" href="glib_005.html#D6" target="_self" title="GLIB_005:def.6">Def10</a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def11</a></i>;<br/><a NAME="E6:30_1_2_2_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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E31">A70</font></i>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th3</a></i>;<br/><a NAME="E8:30_1_2_2_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="glib_005.html#T5" target="_self" title="GLIB_005:th.5">Th4</a></i>;<br/><a NAME="E9:30_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E32">A71</font></i>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="txt" href="#E7:30_1_2_2_1_1_2"><i><font color="Green" title="E31">A70</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/><b>assume </b><a NAME="E1:30_1_2_2_1_1_2_1"/><i><font color="Green" title="E33">A72</font></i>: 
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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="E2:30_1_2_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E34">A73</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_2_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#K4" title="RELSET_1:func.4">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><b>by </b><i><a class="txt" href="#E3:30_1_2_2_1_1_2"><i><font color="Green" title="E29">A68</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_2_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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:30_1_2_2_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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2"><i><font color="Green" title="E25">A62</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E35">A74</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A75</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_2_1_1_1_2"/>
<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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2"><i><font color="Green" title="E27">A65</font></i></a>, <a class="txt" href="#E9:30_1_2_2_1_1_2"><i><font color="Green" title="E32">A71</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E35">A74</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1"><i><font color="Green" title="E34">A73</font></i></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( <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>  &amp; <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>  )
 <b>by </b><i><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></i>;<br/><a NAME="E6:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( not <font color="Maroon" title="c16">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; not <font color="Maroon" title="c17">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A76</font></i>: 
( 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> &amp; 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> )
 <b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><a NAME="E8:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
( 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><b>then reconsider </b><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="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K9" title="REAL_1:func.9">-</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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E10:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E38">A77</font></i>: 
( 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">lenP11</font> &amp; 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">lenP21</font> )
 <b>by </b><i><a class="txt" href="#E7:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E37">A76</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E11:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E39">A78</font></i>: 
( <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> &amp; <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="E12:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E40">A79</font></i>: 
( <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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E13:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E41">A80</font></i>: 
( <font color="Maroon" title="c18">lenP11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font> &amp; <font color="Maroon" title="c19">lenP21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font> )
 <b>by </b><i><a class="txt" href="#E10:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E38">A77</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:30_1_2_2_1_1_2_1_1_1_2"/>
<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>
 <b>by </b><i><a class="txt" href="glib_005.html#E3"><i><font color="Green" title="E3">Tw1</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2"><i><font color="Green" title="E28">A66</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E36">A75</font></i></a></i>;<br/><a NAME="E15:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="ref" href="glib_005.html#T7" target="_self" title="GLIB_005:th.7">Th7</a></i>;<br/><a NAME="E16:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A81</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2_1"><i><font color="Green" title="E34">A73</font></i></a>, <a class="txt" href="#E11:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E39">A78</font></i></a>, <a class="txt" href="#E13:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E41">A80</font></i></a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E18:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E43">A82</font></i>: 
( <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="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c18">lenP11</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font> &amp; <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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font> &amp;  <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> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1"><i><font color="Green" title="E34">A73</font></i></a>, <a class="txt" href="#E13:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E41">A80</font></i></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></i>;<br/><a NAME="E19:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E44">A83</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2"><i><font color="Green" title="E27">A65</font></i></a>, <a class="txt" href="#E5:30_1_2_2_1_1_2"><i><font color="Green" title="E30">A69</font></i></a>, <a class="txt" href="#E18:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E36">A75</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">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> <b> such that </b><br/><a NAME="E21:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E45">A84</font></i>: 
( <font color="Maroon" title="c21">lenP22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">lenP21</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c19">lenP21</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font> &amp; <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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font> &amp;  <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="c21">lenP22</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1"><i><font color="Green" title="E34">A73</font></i></a>, <a class="txt" href="#E13:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E41">A80</font></i></a>, <a class="txt" href="#E16:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E42">A81</font></i></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></i>;<br/><a NAME="E22:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E46">A85</font></i>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2"><i><font color="Green" title="E27">A65</font></i></a>, <a class="txt" href="#E5:30_1_2_2_1_1_2"><i><font color="Green" title="E30">A69</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E45">A84</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E36">A75</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><b>set </b><font color="Maroon" title="c22">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/><b>set </b><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="c21">lenP22</font>;<br/><a NAME="E23:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E47">A86</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">lenP12</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">lenP22</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E24:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E48">A87</font></i>: 
( <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> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E12:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E40">A79</font></i></a>, <a class="txt" href="#E18:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E45">A84</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E25:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A88</font></i>: 
( <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> &amp; <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="c21">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> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1"><i><font color="Green" title="E34">A73</font></i></a>, <a class="txt" href="#E19:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E44">A83</font></i></a>, <a class="txt" href="#E22:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E46">A85</font></i></a>, <a class="txt" href="#E23:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E47">A86</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E26:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E50">A88a</font></i>: 
( <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> &amp; <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="c21">lenP22</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="ref" href="glib_005.html#T2" target="_self" title="GLIB_005:th.2">Th002</a></i>;<br/><div><i><font color="Green" title="E51">A89</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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="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="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="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:30_1_2_2_1_1_2_1_1_1_2_1"/><i><font color="Green" title="E52">A90</font></i>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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="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="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="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_2_1_1_2_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E53">A91</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <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="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> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_2_1_1_1_2_1"/><i><font color="Green" title="E54">A92</font></i>: 
<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="c24">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="c24">n</font>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_2_1_1_1_2_1"><i><font color="Green" title="E52">A90</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E4:30_1_2_2_1_1_2_1_1_1_2_1"/>
<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="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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2_1"><i><font color="Green" title="E53">A91</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
( 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 &amp; <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="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> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:30_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
<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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E55">A93</font></i>: 
<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="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="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="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E8:30_1_2_2_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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></i>;<br/><a NAME="E9:30_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
<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="c16">P1</font>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2_1"><i><font color="Green" title="E53">A91</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:30_1_2_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2_1"><i><font color="Green" title="E53">A91</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:30_1_2_2_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="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="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="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="txt" href="#E3:30_1_2_2_1_1_2_1_1_1_2_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="txt" href="#E7:30_1_2_2_1_1_2_1_1_1_2_1"><i><font color="Green" title="E55">A93</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E52">A94</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_2_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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="c21">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="c21">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="c21">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/><b>assume </b><a NAME="E1:30_1_2_2_1_1_2_1_1_1_2_2"/><i><font color="Green" title="E53">A95</font></i>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c21">lenP22</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c21">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="c21">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_2_1_1_2_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E54">A96</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <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="c21">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_2_1_1_1_2_2"/><i><font color="Green" title="E55">A97</font></i>: 
<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="c21">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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_2_1_1_1_2_2"><i><font color="Green" title="E53">A95</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E4:30_1_2_2_1_1_2_1_1_1_2_2"/>
<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="c21">lenP22</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
( 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 &amp; <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="c21">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:30_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
<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#K4" title="RELSET_1:func.4">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="c21">lenP22</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E56">A98</font></i>: 
<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="c21">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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E8:30_1_2_2_1_1_2_1_1_1_2_2"/>
 <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="c21">lenP22</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">lenP22</font>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></i>;<br/><a NAME="E9:30_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:30_1_2_2_1_1_2_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1_1_1_2_2"><i><font color="Green" title="E54">A96</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:30_1_2_2_1_1_2_1_1_1_2_2"/>
<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="c21">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="c21">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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_2_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="txt" href="#E3:30_1_2_2_1_1_2_1_1_1_2_2"><i><font color="Green" title="E55">A97</font></i></a>, <a class="txt" href="#E7:30_1_2_2_1_1_2_1_1_1_2_2"><i><font color="Green" title="E56">A98</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E29:30_1_2_2_1_1_2_1_1_1_2"/><i><font color="Green" title="E53">A99</font></i>: 
( <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> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E18:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E45">A84</font></i></a></i>;<br/><a NAME="E30:30_1_2_2_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> <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> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E18:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E43">A82</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E45">A84</font></i></a></i>;<br/><a NAME="E31:30_1_2_2_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E54">A100</font></i>: 
( <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> &amp; <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_2_1"><i><font color="Green" title="E34">A73</font></i></a>, <a class="txt" href="#E16:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E42">A81</font></i></a>, <a class="txt" href="#E19:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E44">A83</font></i></a>, <a class="txt" href="#E22:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E46">A85</font></i></a>, <a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="txt" href="#E29:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E53">A99</font></i></a>, <a class="ref" href="glib_001.html#T41" title="GLIB_001:th.41">GLIB_001:41</a></i>;<br/><i><font color="Green" title="E55">A101</font></i>: <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>
<b>by </b><i><a class="txt" href="#E23:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E47">A86</font></i></a>, <a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon" title="c16">P1</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></i>
;<br/><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="c21">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="c21">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>
<b>by </b><i><a class="txt" href="#E23:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E47">A86</font></i></a>, <a class="txt" href="#E24:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E48">A87</font></i></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon" title="c17">P2</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></i>
;<br/><b>hence </b><a NAME="E34:30_1_2_2_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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2"><i><font color="Green" title="E25">A62</font></i></a>, <a class="txt" href="#E5:30_1_2_2_1_1_2"><i><font color="Green" title="E30">A69</font></i></a>, <a class="txt" href="#E25:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E49">A88</font></i></a>, <a class="txt" href="#E26:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E50">A88a</font></i></a>, <a class="txt" href="#E27:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E51">A89</font></i></a>, <a class="txt" href="#E28:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E52">A94</font></i></a>, <a class="txt" href="#E31:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E54">A100</font></i></a>, <a class="txt" href="#E32:30_1_2_2_1_1_2_1_1_1_2"><i><font color="Green" title="E55">A101</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:30_1_2_2_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E11:30_1_2_2_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_3"><b>suppose </b></a><a NAME="E1:30_1_2_2_1_1_3"/><i><font color="Green" title="E27">A102</font></i>: 
(  <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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_3"/><b>then </b><i><font color="Green" title="E28">A103</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2"><i><font color="Green" title="E26">A63</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def14</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_3"/><i><font color="Green" title="E29">A105</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3"><i><font color="Green" title="E28">A103</font></i></a>, <a class="txt" href="glib_005.html#E1"><i><font color="Green" title="E1">Tw0</font></i></a></i>;<br/><a NAME="E4:30_1_2_2_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> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3"><i><font color="Green" title="E27">A102</font></i></a>, <a class="ref" href="glib_005.html#D9" target="_self" title="GLIB_005:def.9">Def13</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_3"/><b>then </b><i><font color="Green" title="E30">A106</font></i>: 
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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3"><i><font color="Green" title="E27">A102</font></i></a>, <a class="ref" href="glib_005.html#D6" target="_self" title="GLIB_005:def.6">Def10</a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def11</a></i>;<br/><a NAME="E6:30_1_2_2_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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_3"/><b>then </b><i><font color="Green" title="E31">A107</font></i>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th3</a></i>;<br/><a NAME="E8:30_1_2_2_1_1_3"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="glib_005.html#T5" target="_self" title="GLIB_005:th.5">Th4</a></i>;<br/><a NAME="E9:30_1_2_2_1_1_3"/><b>then </b><i><font color="Green" title="E32">A108</font></i>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="txt" href="#E7:30_1_2_2_1_1_3"><i><font color="Green" title="E31">A107</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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/><b>assume </b><a NAME="E1:30_1_2_2_1_1_3_1"/><i><font color="Green" title="E33">A109</font></i>: 
( <font color="Maroon" title="c15">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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="E2:30_1_2_2_1_1_3_1"/><b>then </b><i><font color="Green" title="E34">A110</font></i>: 
( <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> &amp; <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> &amp; <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> &amp; <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> )
 <b>by </b><i><a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_3_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_2_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#K4" title="RELSET_1:func.4">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><b>by </b><i><a class="txt" href="#E3:30_1_2_2_1_1_3"><i><font color="Green" title="E29">A105</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_2_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#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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"><b>hence </b><a NAME="E2:30_1_2_2_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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2"><i><font color="Green" title="E25">A62</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E35">A111</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E36">A112</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_3_1_1_1_2"/>
<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>
 <b>by </b><i><a class="txt" href="#E5:30_1_2_2_1_1_3"><i><font color="Green" title="E30">A106</font></i></a>, <a class="txt" href="#E9:30_1_2_2_1_1_3"><i><font color="Green" title="E32">A108</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E35">A111</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( <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>  &amp; <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>  )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( <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>  &amp; <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>  )
 <b>by </b><i><a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></i>;<br/><a NAME="E6:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( not <font color="Maroon" title="c16">P1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; not <font color="Maroon" title="c17">P2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">A113</font></i>: 
( 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> &amp; 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> )
 <b>by </b><i><a class="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/><a NAME="E8:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
( 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><b>then reconsider </b><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="real_1.html#K9" title="REAL_1:func.9">-</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="real_1.html#K9" title="REAL_1:func.9">-</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>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E10:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E38">A114</font></i>: 
( 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">lenP11</font> &amp; 3 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">lenP21</font> )
 <b>by </b><i><a class="txt" href="#E7:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E37">A113</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E11:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E39">A115</font></i>: 
( <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> &amp; <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="E12:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E40">A116</font></i>: 
( <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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <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="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E13:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E41">A117</font></i>: 
( <font color="Maroon" title="c18">lenP11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font> &amp; <font color="Maroon" title="c19">lenP21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font> )
 <b>by </b><i><a class="txt" href="#E10:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E38">A114</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E14:30_1_2_2_1_1_3_1_1_1_2"/>
<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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3"><i><font color="Green" title="E28">A103</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E36">A112</font></i></a>, <a class="txt" href="glib_005.html#E3"><i><font color="Green" title="E3">Tw1</font></i></a></i>;<br/><a NAME="E15:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="ref" href="glib_005.html#T7" target="_self" title="GLIB_005:th.7">Th7</a></i>;<br/><a NAME="E16:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E42">A118</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_3_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="txt" href="#E11:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E39">A115</font></i></a>, <a class="txt" href="#E13:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E41">A117</font></i></a></i>;<br/><b>then consider </b><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> <b> such that </b><br/><a NAME="E18:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E43">A119</font></i>: 
( <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="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c18">lenP11</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font> &amp; <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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font> &amp;  <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> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="txt" href="#E13:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E41">A117</font></i></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></i>;<br/><a NAME="E19:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E44">A120</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3"><i><font color="Green" title="E27">A102</font></i></a>, <a class="txt" href="#E5:30_1_2_2_1_1_3"><i><font color="Green" title="E30">A106</font></i></a>, <a class="txt" href="#E18:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E43">A119</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E36">A112</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">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> <b> such that </b><br/><a NAME="E21:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E45">A121</font></i>: 
( <font color="Maroon" title="c21">lenP22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">lenP21</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 &amp; <font color="Maroon" title="c19">lenP21</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font> &amp; <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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font> &amp;  <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="c21">lenP22</font>,<font color="Maroon" title="c15">v</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="txt" href="#E13:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E41">A117</font></i></a>, <a class="txt" href="#E16:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E42">A118</font></i></a>, <a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></i>;<br/><a NAME="E22:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E46">A122</font></i>: 
<font color="Maroon" title="c17">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3"><i><font color="Green" title="E27">A102</font></i></a>, <a class="txt" href="#E5:30_1_2_2_1_1_3"><i><font color="Green" title="E30">A106</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E45">A121</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E36">A112</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><b>set </b><font color="Maroon" title="c22">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/><b>set </b><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="c21">lenP22</font>;<br/><a NAME="E23:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E47">A123</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">lenP12</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">lenP22</font> )
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E24:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E48">A124</font></i>: 
( <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> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E12:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E40">A116</font></i></a>, <a class="txt" href="#E18:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E43">A119</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E45">A121</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E25:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A125</font></i>: 
( <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> &amp; <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="c21">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> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="txt" href="#E19:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E44">A120</font></i></a>, <a class="txt" href="#E22:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E46">A122</font></i></a>, <a class="txt" href="#E23:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E47">A123</font></i></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></i>;<br/><a NAME="E26:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E50">A125a</font></i>: 
( <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> &amp; <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="c21">lenP22</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> )
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="ref" href="glib_005.html#T2" target="_self" title="GLIB_005:th.2">Th002</a></i>;<br/><div><i><font color="Green" title="E51">A126</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_3_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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="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="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="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E1:30_1_2_2_1_1_3_1_1_1_2_1"/><i><font color="Green" title="E52">A127</font></i>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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:()"><i><font color="Red">::  thesis: </font></i></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="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="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="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_2_1_1_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E53">A128</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <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="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> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_3_1_1_1_2_1"/><i><font color="Green" title="E54">A129</font></i>: 
<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="c24">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="c24">n</font>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_3_1_1_1_2_1"><i><font color="Green" title="E52">A127</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E4:30_1_2_2_1_1_3_1_1_1_2_1"/>
<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="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>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2_1"><i><font color="Green" title="E53">A128</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
( 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 &amp; <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="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> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:30_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
<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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E55">A130</font></i>: 
<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="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="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="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E8:30_1_2_2_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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></i>;<br/><a NAME="E9:30_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
<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="c16">P1</font>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2_1"><i><font color="Green" title="E53">A128</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:30_1_2_2_1_1_3_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">P1</font>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2_1"><i><font color="Green" title="E53">A128</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:30_1_2_2_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="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="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="c24">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="txt" href="#E3:30_1_2_2_1_1_3_1_1_1_2_1"><i><font color="Green" title="E54">A129</font></i></a>, <a class="txt" href="#E7:30_1_2_2_1_1_3_1_1_1_2_1"><i><font color="Green" title="E55">A130</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><i><font color="Green" title="E52">A131</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_3_1_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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="c21">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="c21">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="c21">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/><b>assume </b><a NAME="E1:30_1_2_2_1_1_3_1_1_1_2_2"/><i><font color="Green" title="E53">A132</font></i>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c21">lenP22</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c21">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="c21">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_2_1_1_3_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E54">A133</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">n</font> &amp; <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="c21">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:30_1_2_2_1_1_3_1_1_1_2_2"/><i><font color="Green" title="E55">A134</font></i>: 
<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="c21">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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_3_1_1_1_2_2"><i><font color="Green" title="E53">A132</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E4:30_1_2_2_1_1_3_1_1_1_2_2"/>
<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="c21">lenP22</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2_2"><i><font color="Green" title="E54">A133</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:30_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
( 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 &amp; <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="c21">lenP22</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E6:30_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
<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#K4" title="RELSET_1:func.4">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="c21">lenP22</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:30_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E56">A135</font></i>: 
<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="c21">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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></i>;<br/><a NAME="E8:30_1_2_2_1_1_3_1_1_1_2_2"/>
 <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="c21">lenP22</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">lenP22</font>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="ref" href="glib_001.html#T46" title="GLIB_001:th.46">GLIB_001:46</a></i>;<br/><a NAME="E9:30_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2_2"><i><font color="Green" title="E54">A133</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:30_1_2_2_1_1_3_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">P2</font>
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1_1_1_2_2"><i><font color="Green" title="E54">A133</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>hence </b><a NAME="E11:30_1_2_2_1_1_3_1_1_1_2_2"/>
<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="c21">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="c21">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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2_1_1_3_1"><i><font color="Green" title="E33">A109</font></i></a>, <a class="txt" href="#E3:30_1_2_2_1_1_3_1_1_1_2_2"><i><font color="Green" title="E55">A134</font></i></a>, <a class="txt" href="#E7:30_1_2_2_1_1_3_1_1_1_2_2"><i><font color="Green" title="E56">A135</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E29:30_1_2_2_1_1_3_1_1_1_2"/><i><font color="Green" title="E53">A136</font></i>: 
( <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> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E18:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E43">A119</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E45">A121</font></i></a></i>;<br/><a NAME="E30:30_1_2_2_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> <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> &amp; <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E18:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E43">A119</font></i></a>, <a class="txt" href="#E21:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E45">A121</font></i></a></i>;<br/><a NAME="E31:30_1_2_2_1_1_3_1_1_1_2"/><b>then </b><i><font color="Green" title="E54">A137</font></i>: 
( <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> &amp; <font color="Maroon" title="c17">P2</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <font color="Maroon" title="c21">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> )
 <b>by </b><i><a class="txt" href="#E2:30_1_2_2_1_1_3_1"><i><font color="Green" title="E34">A110</font></i></a>, <a class="txt" href="#E16:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E42">A118</font></i></a>, <a class="txt" href="#E19:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E44">A120</font></i></a>, <a class="txt" href="#E22:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E46">A122</font></i></a>, <a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="txt" href="#E29:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E53">A136</font></i></a>, <a class="ref" href="glib_001.html#T41" title="GLIB_001:th.41">GLIB_001:41</a></i>;<br/><i><font color="Green" title="E55">A138</font></i>: <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>
<b>by </b><i><a class="txt" href="#E23:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E47">A123</font></i></a>, <a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon" title="c16">P1</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></i>
;<br/><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="c21">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="c21">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>
<b>by </b><i><a class="txt" href="#E23:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E47">A123</font></i></a>, <a class="txt" href="#E24:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E48">A124</font></i></a>, <a class="ref" href="glib_001.html#T39" title="GLIB_001:th.39">GLIB_001:39</a></i>
<br/>.= 
<font color="Maroon" title="c17">P2</font>
<b>by </b><i><a class="ref" href="glib_001.html#T40" title="GLIB_001:th.40">GLIB_001:40</a></i>
;<br/><b>hence </b><a NAME="E34:30_1_2_2_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>
 <b>by </b><i><a class="txt" href="#E1:30_1_2_2"><i><font color="Green" title="E25">A62</font></i></a>, <a class="txt" href="#E1:30_1_2_2_1_1_3"><i><font color="Green" title="E27">A102</font></i></a>, <a class="txt" href="#E25:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E49">A125</font></i></a>, <a class="txt" href="#E26:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E50">A125a</font></i></a>, <a class="txt" href="#E27:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E51">A126</font></i></a>, <a class="txt" href="#E28:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E52">A131</font></i></a>, <a class="txt" href="#E31:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E54">A137</font></i></a>, <a class="txt" href="#E32:30_1_2_2_1_1_3_1_1_1_2"><i><font color="Green" title="E55">A138</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:30_1_2_2_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E11:30_1_2_2_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:30_1_2_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:30_1_2"/><b>then </b><i><font color="Green" title="E25">A139</font></i>: 
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/>
<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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:30_1_2"><i><font color="Green" title="E24">A61</font></i></a>, <a class="txt" href="#E4:30_1_2"><i><font color="Green" title="E25">A139</font></i></a>);<br/></i>
<b>hence </b><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#K4" title="RELSET_1:func.4">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="c5">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="c5">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">IT1</font> holds <br/><font color="Maroon" title="c5">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="c5">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="c6">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="c6">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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">IT2</font> holds <br/><font color="Maroon" title="c6">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="c6">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="c5">IT1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c6">IT2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K4" title="RELSET_1:func.4">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="c5">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="c6">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="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> )</span><br/>

<b>thus </b><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#K4" title="RELSET_1:func.4">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="c5">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="c6">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="c5">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">IT2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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