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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:14"/>
<font color="Maroon">c<sub>3</sub></font> <a href="jgraph_1.html#R1" target="_self">is_Shortcut_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="graph_2.html#M1">FinSubsequence</a> of  <a href="jgraph_1.html#K4" target="_self">PairF</a> <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="graph_2.html#M1">FinSubsequence</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font>, <font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:14"/><i><font color="Green">E12</font></i>: 
(  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="finseq_1.html#K15">Seq</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="graph_4.html#R3">is_oriented_vertex_seq_of</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="jgraph_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E4:14"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></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<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="graph_4.html#D5">GRAPH_4:def 5</a></i>;<br/>
<a NAME="E5:14"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="graph_2.html#D5">GRAPH_2:def 5</a></i>;<br/>
<a NAME="E6:14"/><b>then </b><i><font color="Green">E14</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<a NAME="E7:14"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E3:14"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#D14">FINSEQ_1:def 14</a></i>;<br/>
<a NAME="E8:14"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T71">FINSEQ_1:71</a></i>;<br/>
<a NAME="E9:14"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E7:14"><i><font color="Green">E15</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="jgraph_1.html#E6:14"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E12:14"/><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jgraph_1.html#E6:14"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E18</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jgraph_1.html#E9:14"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="jgraph_1.html#E12:14"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>
;<br/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>9</sub></font> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>
;<br/>
<b>hence </b><a NAME="E15:14"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jgraph_1.html#E3:14"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jgraph_1.html#E4:14"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jgraph_1.html#E6:14"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jgraph_1.html#E13:14"><i><font color="Green">E18</font></i></a>, <a class="ref" href="card_1.html#T80">CARD_1:80</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>


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