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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#V9" title="GLIB_003:attr.9">real-elabeled</a> <a href="glib_003.html#NM4" title="GLIB_003:NM.4">WEGraph</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">source</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Olive" title="b1">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Olive" title="b1">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v</font></span><br/><b>let </b><font color="Maroon" title="c2">source</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</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"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font></span><br/><b>let </b><font color="Maroon" title="c3">n</font> be    <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> ; <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">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font></span><br/><b>let </b><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  implies <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> )</span><br/>







<b>set </b><font color="Maroon" title="c5">CS</font> =  <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font>;<br/>
<b>set </b><font color="Maroon" title="c6">L</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> ;<br/>
<b>set </b><font color="Maroon" title="c7">GL</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">GL1</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ; <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"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  holds <br/><span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>;<br/>
<a NAME="E2:44"/><i><font color="Green" title="E23">A2</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="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">k</font> be    <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> ; <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="c9">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><b>set </b><font color="Maroon" title="c10">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><a NAME="E2:44_1"/><i><font color="Green" title="E25">A4</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></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>  <a href="glib_005.html#K4" title="GLIB_005:func.4">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def15</a></i>;<br/><b>set </b><font color="Maroon" title="c12">Next</font> =  <a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c13">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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_1_1_1"/>
(  <a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  ) or (  <a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1"/>
 <a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:44_1_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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></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> <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:44_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44_1"><i><font color="Green" title="E24">A3</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="44_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_2"/><i><font color="Green" title="E26">A5</font></i>: 
(  <a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:44_1_1_1_2"/><b>then </b><i><font color="Green" title="E27">A6</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:44_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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"> ( <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  implies <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> )</span><br/><b>assume </b><a NAME="E1:44_1_1_1_2_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ; <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"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font></span><br/><a NAME="E2:44_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/><a NAME="E3:44_1_1_1_2_1"/>
<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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th4</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:44_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c14">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44_1_1_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="glib_005.html#E1:44_1_1_1_2_1"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:44_1_1_1_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="glib_005.html#E2:44_1_1_1_2_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="glib_003.html#T50" title="GLIB_003:th.50">GLIB_003:50</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><b>hence </b><a NAME="E4:44_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="44_1_1_1_3"><b>suppose </b></a><a NAME="E1:44_1_1_1_3"/><i><font color="Green" title="E26">A9</font></i>: 
(  <a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</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="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:44_1_1_1_3"/><b>then </b><i><font color="Green" title="E27">A10</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</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> <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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:44_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="glib_005.html#D14" target="_self" title="GLIB_005:def.14">Def14</a></i>;<br/><a NAME="E3:44_1_1_1_3"/><i><font color="Green" title="E28">A11</font></i>: 
(  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_005.html#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_labeling_in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> 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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_labeling_in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44_1_1_1_3"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="glib_005.html#D13" target="_self" title="GLIB_005:def.13">Def13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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"> ( <font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  implies <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> )</span><br/><b>assume </b><a NAME="E1:44_1_1_1_3_1"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c14">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 ; <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"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font></span><br/><a NAME="E2:44_1_1_1_3_1"/><b>then </b><i><font color="Green" title="E30">A13</font></i>: 
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/><a NAME="E3:44_1_1_1_3_1"/>
<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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a> 
 <b>by </b><i><a class="ref" href="glib_005.html#T4" target="_self" title="GLIB_005:th.4">Th4</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E4:44_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c14">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></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#K3" title="GLIB_005:func.3">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44_1_1_1_3"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="glib_005.html#E3:44_1_1_1_3"><i><font color="Green" title="E28">A11</font></i></a>, <a class="txt" href="glib_005.html#E1:44_1_1_1_3_1"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="glib_005.html#D10" target="_self" title="GLIB_005:def.10">Def10</a>, <a class="ref" href="glib_005.html#D11" target="_self" title="GLIB_005:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">v</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E2:44_1_1_1_3"><i><font color="Green" title="E27">A10</font></i></a>, <a class="txt" href="glib_005.html#E2:44_1_1_1_3_1"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="glib_003.html#T50" title="GLIB_003:th.50">GLIB_003:50</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><b>hence </b><a NAME="E5:44_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="E4:44_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="E4:44"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
;<br/>
<a NAME="E5:44"/><i><font color="Green" title="E25">A15</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_005.html#E2:44"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="glib_005.html#E4:44"><i><font color="Green" title="E24">A14</font></i></a>);<br/></i>
<a NAME="E6:44"/><i><font color="Green" title="E26">A16</font></i>: 
 <a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font> is <a href="glib_000.html#V13" title="GLIB_000:attr.13">halting</a>
 
<b>by </b><i><a class="ref" href="glib_005.html#T6" target="_self" title="GLIB_005:th.6">Th6</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>means </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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span>;<br/>
<a NAME="E7:44"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E28">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><b>set </b><font color="Maroon" title="c10">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>assume </b><a NAME="E1:44_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</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>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
 <a href="glib_005.html#K4" title="GLIB_005:func.4">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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def15</a></i>
;<br/><b>hence </b><a NAME="E3:44_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_005.html#E6:44"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="glib_000.html#D57" title="GLIB_000:def.57">GLIB_000:def 57</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="E9:44"/><i><font color="Green" title="E29">A19</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="glib_005.html#E7:44"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="glib_005.html#E8:44"><i><font color="Green" title="E28">A18</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:44_3_1"/>
( <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  or <span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_3_1_1"><b>suppose </b></a><a NAME="E1:44_3_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 ; <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"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:44_3_1_1"/><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font> as    <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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:44_3_1_1"/>
<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 <b>by </b><i><a class="txt" href="glib_005.html#E3:44_3_1_1"><i><font color="Green" title="E30">A20</font></i></a></i>;<br/><b>hence </b><a NAME="E6:44_3_1_1"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E1:44"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="glib_005.html#E5:44"><i><font color="Green" title="E25">A15</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="44_3_1_2"><b>suppose </b></a><a NAME="E1:44_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K5" title="GLIB_005:func.5">AP:CompSeq</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">n</font>
 ; <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"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:44_3_1_2"/><i><font color="Green" title="E30">A21</font></i>: 
<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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">k</font> = <font color="Maroon" title="c9">k</font> as    <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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:44_3_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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E3:44_3_1_2"><i><font color="Green" title="E30">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E6:44_3_1_2"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="txt" href="glib_005.html#E9:44"><i><font color="Green" title="E29">A19</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="E11:44"/>
<span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</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="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K7" title="GLIB_003:func.7">the_VLabel_of</a> <span class="p2">(<span class="default"><a href="glib_005.html#K6" title="GLIB_005:func.6">AP:FindAugPath</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">source</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</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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