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<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="graph_1.html#NM1">Graph</a>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="graph_1.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="graph_1.html#E15"><i><font color="Green">Lemma12</font></i></a></i>;<br/>
<a NAME="E2:19"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:19_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a><br/>

<a NAME="E2:19_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:19_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E4:19_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> or the <a href="graph_1.html#U3" target="_self">Source</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="graph_1.html#U4" target="_self">Target</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
 <a href="xboole_0.html#K1">{}</a>  is  <a href="graph_1.html#V8" target="_self">oriented</a>  <a href="graph_1.html#M2" target="_self">Chain</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="graph_1.html#D12" target="_self">Def12</a></i>;<br/>


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