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<div class="add">

<b>consider </b><font color="Maroon">q</font> being  <a href="xboole_0.html#V1">empty</a> <a href="graph_1.html#V8">oriented</a>  <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font>;<br/>
<b>consider </b><font color="Maroon">x</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>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="binarith.html#K11">*&gt;</a></span> as    <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/>
<a NAME="E4:26_1_1"/>
<font color="Maroon">p</font> <a href="graph_4.html#R3" target="_self">is_oriented_vertex_seq_of</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:26_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="graph_2.html#R1">is_vertex_seq_of</a> <font color="Maroon">q</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> ;<br/>
<a NAME="E7:26_1_1"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Olive">m</font> holds <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:26_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:26_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1_1_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:26_1_1"/><b>then </b>
<font color="Maroon">q'</font> is <a href="graph_2.html#V3">simple</a>
 
<b>by </b><i>, <a class="ref" href="graph_2.html#D10">GRAPH_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:26_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="graph_1.html#M2">Chain</a> of <font color="Maroon">G</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="graph_1.html#V8">oriented</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="graph_2.html#V3">simple</a> )
 ;<br/>


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