<?xml version="1.0"?>
<div class="add">

<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> be  non <a href="struct_0.html#V3">empty</a>  <a href="fsm_1.html#L1">FSM</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="fsm_2.html#V3" target="_self">regular</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="fsm_2.html#V1" target="_self">calculating_type</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><a NAME="E2:24"/><i><font color="Green">E14</font></i>: 
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E3:24"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="fsm_2.html#V2" target="_self">accessible</a>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E1:24"><i><font color="Green">E13</font></i></a>, <a class="ref" href="fsm_2.html#D4" target="_self">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:24"/><i><font color="Green">E15</font></i>: 
the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="fsm_1.html#R1">-leads_to</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="fsm_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E5:24"><i><font color="Green">E15</font></i></a>, <a class="ref" href="fsm_1.html#D3">FSM_1:def 3</a></i>;<br/>
<b>assume </b><a NAME="E7:24"/><i><font color="Green">E17</font></i>: 
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>, <font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E9:24"/><i><font color="Green">E18</font></i>: 
 <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E7:24"><i><font color="Green">E17</font></i></a>, <a class="ref" href="fsm_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E10:24"/>
 <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E7:24"><i><font color="Green">E17</font></i></a>, <a class="ref" href="fsm_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E11:24"/><b>then </b><i><font color="Green">E19</font></i>: 
( <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 3 <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="fsm_2.html#E9:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>
<a NAME="E12:24"/><i><font color="Green">E20</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 2 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 2 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E13:24"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
;<br/>
<a NAME="E14:24"/><b>then </b><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3</span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="txt" href="fsm_2.html#E5:24"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fsm_2.html#E11:24"><i><font color="Green">E19</font></i></a>, <a class="ref" href="fsm_1.html#T22">FSM_1:22</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:24_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:24_1_1"/><i><font color="Green">E23</font></i>: 
(  <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>8</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> &amp;  <a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>9</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><span class="p4">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="fsm_2.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:24_1_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:24_1_1"/><i><font color="Green">E25</font></i>: 
( 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; 3 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="fsm_2.html#E12:24"><i><font color="Green">E20</font></i></a></i>;<br/><i><font color="Green">E26</font></i>: <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>8</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 3 = 
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_2.html#E3:24_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_2.html#E6:24"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fsm_2.html#E7:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fsm_2.html#E2:24_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>
;<br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <font color="Maroon">c<sub>9</sub></font>,the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 3 = 
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_2.html#E3:24_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>
<br/>.= 
the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">the <a href="fsm_1.html#U2">InitS</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="fsm_2.html#E6:24"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fsm_2.html#E7:24"><i><font color="Green">E17</font></i></a>, <a class="txt" href="fsm_2.html#E2:24_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>
;<br/><b>hence </b><a NAME="E8:24_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_2.html#E1:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fsm_2.html#E2:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fsm_2.html#E4:24_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fsm_2.html#E5:24_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fsm_2.html#E6:24_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="fsm_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c<sub>11</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E3:24_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <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>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="rewrite1.html#T5">REWRITE1:5</a></i>;<br/><a NAME="E4:24_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E5:24_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>then </b><i><font color="Green">E24</font></i>: <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="fsm_2.html#E3:24_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>
;<br/><a NAME="E7:24_1_2"/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="fsm_2.html#E3:24_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/><a NAME="E8:24_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="fsm_2.html#E5:24_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fsm_2.html#E6:24_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E9:24_1_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="fsm_2.html#E12:24"><i><font color="Green">E20</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E10:24_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <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>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><b>hence </b><a NAME="E11:24_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fsm_2.html#E1:24"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fsm_2.html#E2:24"><i><font color="Green">E14</font></i></a>, <a class="txt" href="fsm_2.html#E14:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="fsm_2.html#E8:24_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fsm_2.html#D1" target="_self">Def1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
