<?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>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:36"/><i><font color="Green">E18</font></i>: 
 <a href="lang1.html#K5" target="_self">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lang1.html#T16" target="_self">Th16</a></i>;<br/>
<b>thus </b><a NAME="E2:36"/>
 <a href="lang1.html#K7" target="_self">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K7" target="_self">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:36_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lang1.html#K5" target="_self">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> )
 ;<br/>
<a NAME="E4:36_1"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lang1.html#E1:36"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lang1.html#E3:36_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="txt" href="lang1.html#E3:36_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E3:36"/><i><font color="Green">E19</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; the <a href="lang1.html#U1" target="_self">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="lang1.html#D10" target="_self">Def10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>;<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="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> 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; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>;<br/>
<a NAME="E4:36"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:36_2"/>
 <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="E2:36_2"/><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="E3:36_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<b>hence </b><a NAME="E4:36_2"/>
(  not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  or <font color="Maroon">c<sub>4</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> )
 <b>by </b><i><a class="ref" href="lang1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:36"/><i><font color="Green">E21</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="36_3"><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="E1:36_3"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> 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; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>assume </b><a NAME="E2:36_3"/><i><font color="Green">E23</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 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a>  )
 ;<br/>

<b>then 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="E4:36_3"/><i><font color="Green">E24</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="ref" href="trees_2.html#T4">TREES_2:4</a></i>;<br/>
<a NAME="E5:36_3"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lang1.html#E2:36_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E6:36_3"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <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> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <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> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K2">rng</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="lang1.html#E4:36_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T50">FINSEQ_1:50</a>, <a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E8:36_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 
<b>by </b><i><a class="txt" href="lang1.html#E6:36_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:36_3"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="lang1.html#E1:36_3"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lang1.html#E4:36_3"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E10:36_3"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="lang1.html#E6:36_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="lang1.html#NM1">Symbol</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="lang1.html#E10:36_3"><i><font color="Green">E27</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E13:36_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p3"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
;<br/>
<a NAME="E14:36_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="lang1.html#U1" target="_self">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lang1.html#E3:36"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E15:36_3"/><b>then </b>
the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lang1.html#R1" target="_self">==&gt;</a> <span class="p1"><a href="lang1.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K4" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="lang1.html#E3:36"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lang1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E16:36_3"/><b>then </b>
<span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R2" target="_self">==&gt;</a> <span class="p1"><a href="lang1.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K4" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="lang1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E17:36_3"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R2" target="_self">==&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K4" target="_self">*&gt;</a></span> &amp; <span class="p1"><a href="lang1.html#K4" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K4" target="_self">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="lang1.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/>
<a NAME="E18:36_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R2" target="_self">==&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="lang1.html#E4:36_3"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E19:36_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="lang1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E20:36_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="lang1.html#E9:36_3"><i><font color="Green">E26</font></i></a>, <a class="ref" href="lang1.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36"/><i><font color="Green">E22</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="lang1.html#E4:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="lang1.html#E5:36"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:36"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:36"/><i><font color="Green">E24</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E10:36"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lang1.html#E3:36"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:36"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E13:36"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E14:36"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="lang1.html#E6:36"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lang1.html#E7:36"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E15:36"/><i><font color="Green">E26</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E16:36"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E17:36"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="lang1.html#U1" target="_self">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lang1.html#E3:36"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E18:36"/><b>then </b>
the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lang1.html#R1" target="_self">==&gt;</a>  <a href="lang1.html#K2" target="_self">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="lang1.html#E3:36"><i><font color="Green">E19</font></i></a>, <a class="ref" href="lang1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E19:36"/><b>then </b>
<span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R2" target="_self">==&gt;</a>  <a href="lang1.html#K2" target="_self">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lang1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E20:36"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R2" target="_self">==&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="lang1.html#E15:36"><i><font color="Green">E26</font></i></a>, <a class="ref" href="lang1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E21:36"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <font color="Maroon">c<sub>6</sub></font> <a href="lang1.html#K1" target="_self">^</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="lang1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E22:36"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="lang1.html#E14:36"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lang1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E23:36"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K7" target="_self">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K13" target="_self">TotalGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lang1.html#E1:36"><i><font color="Green">E18</font></i></a>, <a class="txt" href="lang1.html#E9:36"><i><font color="Green">E24</font></i></a></i>;<br/>


</div>
