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<b>let </b><font color="Maroon">G</font> be   <a href="graph_1.html#NM1">Graph</a>;<br/><b>let </b><font color="Maroon">pe</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v1</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v2</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">v3</font> be    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">p</font> be  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/>











<b>assume </b><a NAME="E1:44"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">p</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font> &amp; <font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1 <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">v2</font>,<font color="Maroon">v3</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">pe</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">FT</font> = the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font>;<br/>
<b>set </b><font color="Maroon">FS</font> = the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font>;<br/>
<a NAME="E2:44"/><i><font color="Green">E45</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:44"/><i><font color="Green">E47</font></i>: 
( the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> &amp; the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">pe</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font> )
 
<b>by </b><i>, <a class="ref" href="graph_4.html#D1">GRAPH_4:def 1</a></i>;<br/>
<a NAME="E4:44"/>
<font color="Maroon">pe</font> is  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">pe</font> as  <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<a NAME="E6:44"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:44"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E8:44"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:44"/><i><font color="Green">E52</font></i>: 
the <a href="graph_1.html#U3">Source</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:44"/>
the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v3</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E11:44"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">pe</font> &amp; <font color="Maroon">q</font> <a href="graph_5.html#R1" target="_self">is_orientedpath_of</a> <font color="Maroon">v1</font>,<font color="Maroon">v3</font> )
 <b>by </b><i><a class="txt" href="graph_5.html#E2:44"><i><font color="Green">E45</font></i></a>, , </i>;<br/>


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