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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="pnproc_1.html#M3" target="_self">Petri_net</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pnproc_1.html#NM5">firing-sequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>3</sub></font> ) holds <br/>(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  );<br/>
<a NAME="E1:65"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:65_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:65_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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="E3:65_1"/><b>then </b>
 <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K6">id</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T41">FUNCT_7:41</a></i>;<br/>
<b>hence </b><a NAME="E4:65_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> holds not <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>2</sub></font> ) ) or (  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  ) )
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:65"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> st  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>3</sub></font> ) holds <br/>(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:65_2"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E3:65_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:65_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E2:65_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="trees_2.html#T4">TREES_2:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="txt" href="pnproc_1.html#E5:65_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_7.html#T25">FUNCT_7:25</a></i>;<br/>
<a NAME="E7:65_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:65_2_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E2:65_2_1"/><b>then </b>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:65_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E2:65_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:65_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:65_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E10:65_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:65_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>;<br/>
<a NAME="E11:65_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:65_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pnproc_1.html#E9:65_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_7.html#T43">FUNCT_7:43</a></i>;<br/>
<a NAME="E12:65_2"/><i><font color="Green">E52</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="pnproc_1.html#D9" target="_self">Def9</a>, <a class="ref" href="pnproc_1.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E13:65_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:65_2_2"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:65_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#T39">FINSEQ_1:39</a></i>;<br/>
<b>assume </b><a NAME="E2:65_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:65_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:65_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pnproc_1.html#E1:65_2_2"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:65_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E5:65_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pnproc_1.html#E2:65_2_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:65_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E4:65_2_2"><i><font color="Green">E55</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:65_2"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E1:65_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="pnproc_1.html#E5:65_2"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E15:65_2"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="pnproc_1.html#E11:65_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="pnproc_1.html#E12:65_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>hence </b><a NAME="E16:65_2"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E12:65_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:65"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="pnproc_1.html#E1:65"><i><font color="Green">E44</font></i></a>, <a class="txt" href="pnproc_1.html#E2:65"><i><font color="Green">E45</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being  <a href="funcop_1.html#V1">Function-yielding</a> <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:65"/><i><font color="Green">E47</font></i>: 
 <a href="pnproc_1.html#K10" target="_self">fire</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="funct_7.html#K4">compose</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a> </span>)</span>
 <b>and </b><br/><a NAME="E6:65"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E7:65"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="pnproc_1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E8:65"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="pnproc_1.html#M2" target="_self">transition</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:65_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:65_3"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</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>4</sub></font></span>)</span> &amp;  <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#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="pnproc_1.html#M4" target="_self">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E4:65_3"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pnproc_1.html#K9" target="_self">fire</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pnproc_1.html#E6:65"><i><font color="Green">E48</font></i></a>, <a class="txt" href="pnproc_1.html#E7:65"><i><font color="Green">E49</font></i></a>, <a class="txt" href="pnproc_1.html#E1:65_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="pnproc_1.html#E2:65_3"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:65"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K10" target="_self">fire</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K10" target="_self">fire</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="txt" href="pnproc_1.html#E3:65"><i><font color="Green">E46</font></i></a>, <a class="txt" href="pnproc_1.html#E5:65"><i><font color="Green">E47</font></i></a>, <a class="txt" href="pnproc_1.html#E6:65"><i><font color="Green">E48</font></i></a></i>;<br/>


</div>
