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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">vs</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">c</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="E1:12"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">vs</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">c</font>
 ;<br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:12"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> <a href="graph_4.html#K2" target="_self">-TVSet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">v</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U1">Vertices</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  ex <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="graph_1.html#U2">Edges</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:12"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4">Target</a> of <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:12"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E11:12"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E12:12"/><b>then </b><i><font color="Green">E49</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>set </b><font color="Maroon">v1</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>;<br/>
<b>set </b><font color="Maroon">v2</font> = <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/>
<a NAME="E13:12"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>;<br/>
<a NAME="E14:12"/>
<font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="graph_4.html#R1" target="_self">orientedly_joins</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>,<font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:12"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E16:12"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> &amp; <font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">vs</font> )
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E17:12"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">vs</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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