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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font>, <font color="Maroon">g</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="jgraph_1.html#R1" target="_self">is_Shortcut_of</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> &amp;  ex <font color="Olive">fc</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of  <a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">f</font> ex <font color="Olive">fvs</font> being   <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">f</font> ex <font color="Olive">sc</font> being <a href="graph_1.html#V8">oriented</a> <a href="graph_2.html#V3">simple</a>  <a href="graph_1.html#M2">Chain</a> of  <a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font> ex <font color="Olive">g1</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span> st <br/>(  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fc</font> <a href="hidden.html#R1">=</a> <font color="Olive">sc</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Olive">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Olive">g1</font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Olive">sc</font> ) )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">fc</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of  <a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">f</font>, <font color="Maroon">fvs</font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">f</font>, <font color="Maroon">sc</font> being  <a href="graph_1.html#V8">oriented</a> <a href="graph_2.html#V3">simple</a>  <a href="graph_1.html#M2">Chain</a> of  <a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font>, <font color="Maroon">g1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E4:17"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">sc</font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">fvs</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g1</font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">sc</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E5:17"/>
<font color="Maroon">sc</font> is <a href="graph_4.html#V1">Simple</a>
 
<b>by </b><i><a class="ref" href="graph_4.html#T20">GRAPH_4:20</a></i>;<br/>
<b>then consider </b><font color="Maroon">vs</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="graph_1.html#U1">Vertices</a> of <span class="p1">(<span class="default"><a href="jgraph_1.html#K1" target="_self">PGraph</a> <font color="Maroon">X</font></span>)</span><b> such that </b><br/><a NAME="E7:17"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">vs</font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">sc</font> &amp; ( for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">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">vs</font> &amp; <font color="Maroon">vs</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</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">vs</font> ) ) )
 <b>by </b><i><a class="ref" href="graph_4.html#D7">GRAPH_4:def 7</a></i>;<br/>
<a NAME="E8:17"/><i><font color="Green">E38</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> holds <br/><font color="Maroon">sc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> <a href="graph_4.html#R1">orientedly_joins</a> <font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font>,<font color="Maroon">g1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/>
<a NAME="E9:17"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:17"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g1</font>
 
<b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E11:17"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E12:17"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">sc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E13:17"/><b>then </b>
<font color="Maroon">sc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E14:17"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vs</font>
 
<b>by </b><i>, , <a class="ref" href="graph_4.html#T11">GRAPH_4:11</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <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="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E15:17"/><i><font color="Green">E40</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">g</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E16:17"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E18:17"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">i2</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E20:17"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:17"/><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>assume </b><a NAME="E22:17"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> or <font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>suppose </b></a><a NAME="E1:17_1_1"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1"/><b>then </b>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:17_1_1"/><b>then </b>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , , , </i>;<br/><b>hence </b><a NAME="E4:17_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><b>suppose </b></a><a NAME="E1:17_1_2"/>
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font>
 ;<br/><div class="add"><a NAME="E2:17_1_2"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , , , , </i>;<br/><b>hence </b><a NAME="E3:17_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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