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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</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">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b2">e</font></span>)</span> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">W</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c2">W</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</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">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Olive" title="b1">e</font></span>)</span> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c3">e</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="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font></span>)</span> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/>





<b>set </b><font color="Maroon" title="c4">W2</font> = <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font>;<br/>
<b>set </b><font color="Maroon" title="c5">WA</font> = <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"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c3">e</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K30" title="GLIB_000:func.30">.adj</a> <font color="Maroon" title="c3">e</font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:164"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</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"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font></span>)</span> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/>

<a NAME="E2:164"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c3">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K30" title="GLIB_000:func.30">.adj</a> <font color="Maroon" title="c3">e</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_001.html#D13" title="GLIB_001:def.13">GLIB_001:def 13</a></i>;<br/>
<a NAME="E3:164"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K30" title="GLIB_000:func.30">.adj</a> <font color="Maroon" title="c3">e</font>,<font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="#E1:164"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="glib_000.html#T70" title="GLIB_000:th.70">GLIB_000:70</a></i>;<br/>
<a NAME="E4:164"/><b>then </b>
<font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c3">e</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K30" title="GLIB_000:func.30">.adj</a> <font color="Maroon" title="c3">e</font></span>)</span></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 
<b>by </b><i><a class="ref" href="glib_001.html#T16" title="GLIB_001:th.16">GLIB_001:16</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K10" title="GLIB_001:func.10">.addEdge</a> <font color="Maroon" title="c3">e</font></span>)</span> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_001.html#K2" title="GLIB_001:func.2">.walkOf</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span>,<font color="Maroon" title="c3">e</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span> <a href="glib_000.html#K30" title="GLIB_000:func.30">.adj</a> <font color="Maroon" title="c3">e</font></span>)</span></span>)</span> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span>
<b>by </b><i><a class="txt" href="#E2:164"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="glib_003.html#T23" target="_self" title="GLIB_003:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">W</font> <a href="glib_003.html#K8" title="GLIB_003:func.8">.weightSeq()</a> </span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">e</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E3:164"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="glib_003.html#T21" target="_self" title="GLIB_003:th.21">Th21</a></i>
;<br/> <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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