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









<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>;<br/>
<a NAME="E1:4"/><i><font color="Green">E3</font></i>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E2:4"/><i><font color="Green">E4</font></i>: 
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/>
<a NAME="E3:4"/><i><font color="Green">E5</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E4:4"/>
 <a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="fsm_2.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:4"/><b>then </b>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>4</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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E6:4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font>,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="fsm_1.html#R1">-leads_to</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>4</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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E3:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="fsm_1.html#D3">FSM_1:def 3</a></i>;<br/>
<a NAME="E7:4"/><b>then </b><i><font color="Green">E6</font></i>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>4</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>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="txt" href="fsm_2.html#E3:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="fsm_1.html#T21">FSM_1:21</a></i>;<br/>
<a NAME="E8:4"/><i><font color="Green">E7</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <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="E9:4"/>
( 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="fsm_1.html#NM1">State</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E11:4"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 2 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="fsm_1.html#K1">-succ_of</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>;<br/>
<a NAME="E12:4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E11:4"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E13:4"/><b>then </b><i><font color="Green">E9</font></i>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>4</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>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="fsm_2.html#E7:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fsm_2.html#E11:4"><i><font color="Green">E8</font></i></a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="fsm_1.html#D2">FSM_1:def 2</a></i>
<br/>.= 
3
<b>by </b><i><a class="txt" href="fsm_2.html#E8:4"><i><font color="Green">E7</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E15:4"/>
 <a href="fsm_2.html#NK1" target="_self">GEN</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="finseq_4.html#K2">*&gt;</a></span>,<font color="Maroon">c<sub>5</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>5</sub></font>,<span class="p2">(<span class="default">the <a href="fsm_1.html#U1">Tran</a> of <font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</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>4</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>4</sub></font> <a href="funct_2.html#K3">.</a> <span class="p5"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<font color="Maroon">c<sub>3</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#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="fsm_2.html#E7:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fsm_2.html#E13:4"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T62">FINSEQ_1:62</a></i>;<br/>


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